LMFDB - L-function 14-464e7-1.1-c5e7-0-0

Dirichlet series

L(s) = 1

− 26·3^-s + 32·5^-s − 184·7^-s − 10·9^-s − 1.10e3·11^-s + 408·13^-s − 832·15^-s − 874·17^-s − 4.28e3·19^-s + 4.78e3·21^-s + 4.53e3·23^-s − 7.66e3·25^-s + 6.24e3·27^-s + 5.88e3·29^-s − 7.79e3·31^-s + 2.87e4·33^-s − 5.88e3·35^-s + 5.08e3·37^-s − 1.06e4·39^-s + 1.98e4·41^-s − 1.94e4·43^-s − 320·45^-s − 1.42e4·47^-s − 2.26e4·49^-s + 2.27e4·51^-s − 5.86e4·53^-s − 3.53e4·55^-s + ⋯

L(s) = 1

− 1.66·3^-s + 0.572·5^-s − 1.41·7^-s − 0.0411·9^-s − 2.75·11^-s + 0.669·13^-s − 0.954·15^-s − 0.733·17^-s − 2.72·19^-s + 2.36·21^-s + 1.78·23^-s − 2.45·25^-s + 1.64·27^-s + 1.29·29^-s − 1.45·31^-s + 4.59·33^-s − 0.812·35^-s + 0.610·37^-s − 1.11·39^-s + 1.84·41^-s − 1.60·43^-s − 0.0235·45^-s − 0.942·47^-s − 1.34·49^-s + 1.22·51^-s − 2.86·53^-s − 1.57·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 29^{7}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$14$
Conductor:	$2^{28} \cdot 29^{7}$
Sign:	$-1$
Analytic conductor:	$1.26400\times 10^{13}$
Root analytic conductor:	$8.62659$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$7$
Selberg data:	$(14,\ 2^{28} \cdot 29^{7} ,\ ( \ : [5/2]^{7} ),\ -1 )$

Particular Values

$L(3)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	29	$ ( 1 - p^{2} T )^{7} $
good	3	$ 1 + 26 T + 686 T^{2} + 11852 T^{3} + 287083 T^{4} + 5198968 T^{5} + 28114595 p T^{6} + 134680040 p^{2} T^{7} + 28114595 p^{6} T^{8} + 5198968 p^{10} T^{9} + 287083 p^{15} T^{10} + 11852 p^{20} T^{11} + 686 p^{25} T^{12} + 26 p^{30} T^{13} + p^{35} T^{14} $
	5	$ 1 - 32 T + 8686 T^{2} - 28766 p T^{3} + 10628299 p T^{4} - 867911724 T^{5} + 216595986043 T^{6} - 2190219197314 T^{7} + 216595986043 p^{5} T^{8} - 867911724 p^{10} T^{9} + 10628299 p^{16} T^{10} - 28766 p^{21} T^{11} + 8686 p^{25} T^{12} - 32 p^{30} T^{13} + p^{35} T^{14} $
	7	$ 1 + 184 T + 56537 T^{2} + 6550512 T^{3} + 1475543389 T^{4} + 153397502280 T^{5} + 30359255728661 T^{6} + 2840842475508000 T^{7} + 30359255728661 p^{5} T^{8} + 153397502280 p^{10} T^{9} + 1475543389 p^{15} T^{10} + 6550512 p^{20} T^{11} + 56537 p^{25} T^{12} + 184 p^{30} T^{13} + p^{35} T^{14} $
	11	$ 1 + 1106 T + 1142534 T^{2} + 67722540 p T^{3} + 483650332067 T^{4} + 239055935973032 T^{5} + 10747729575295051 p T^{6} + 389095375019586912 p^{2} T^{7} + 10747729575295051 p^{6} T^{8} + 239055935973032 p^{10} T^{9} + 483650332067 p^{15} T^{10} + 67722540 p^{21} T^{11} + 1142534 p^{25} T^{12} + 1106 p^{30} T^{13} + p^{35} T^{14} $
	13	$ 1 - 408 T + 1499278 T^{2} - 459590126 T^{3} + 844969463063 T^{4} - 182571255148732 T^{5} + 266089691249902643 T^{6} - 50615732075027218682 T^{7} + 266089691249902643 p^{5} T^{8} - 182571255148732 p^{10} T^{9} + 844969463063 p^{15} T^{10} - 459590126 p^{20} T^{11} + 1499278 p^{25} T^{12} - 408 p^{30} T^{13} + p^{35} T^{14} $
	17	$ 1 + 874 T + 3461027 T^{2} + 3565345812 T^{3} + 8096908141505 T^{4} + 8791339726729270 T^{5} + 14877189201463598459 T^{6} + $$13\!\cdots\!40$$ T^{7} + 14877189201463598459 p^{5} T^{8} + 8791339726729270 p^{10} T^{9} + 8096908141505 p^{15} T^{10} + 3565345812 p^{20} T^{11} + 3461027 p^{25} T^{12} + 874 p^{30} T^{13} + p^{35} T^{14} $
	19	$ 1 + 4288 T + 15892769 T^{2} + 44193901424 T^{3} + 108123320266529 T^{4} + 11780519511376640 p T^{5} + $$42\!\cdots\!17$$ T^{6} + $$36\!\cdots\!56$$ p T^{7} + $$42\!\cdots\!17$$ p^{5} T^{8} + 11780519511376640 p^{11} T^{9} + 108123320266529 p^{15} T^{10} + 44193901424 p^{20} T^{11} + 15892769 p^{25} T^{12} + 4288 p^{30} T^{13} + p^{35} T^{14} $
	23	$ 1 - 4532 T + 47162205 T^{2} - 167997820744 T^{3} + 939675247549225 T^{4} - 2635601398175683724 T^{5} + $$10\!\cdots\!13$$ T^{6} - $$22\!\cdots\!44$$ T^{7} + $$10\!\cdots\!13$$ p^{5} T^{8} - 2635601398175683724 p^{10} T^{9} + 939675247549225 p^{15} T^{10} - 167997820744 p^{20} T^{11} + 47162205 p^{25} T^{12} - 4532 p^{30} T^{13} + p^{35} T^{14} $
	31	$ 1 + 7794 T + 125103218 T^{2} + 940790849968 T^{3} + 8446944371681359 T^{4} + 52568423948451215520 T^{5} + $$36\!\cdots\!93$$ T^{6} + $$18\!\cdots\!44$$ T^{7} + $$36\!\cdots\!93$$ p^{5} T^{8} + 52568423948451215520 p^{10} T^{9} + 8446944371681359 p^{15} T^{10} + 940790849968 p^{20} T^{11} + 125103218 p^{25} T^{12} + 7794 p^{30} T^{13} + p^{35} T^{14} $
	37	$ 1 - 5086 T + 211423091 T^{2} - 1670989337316 T^{3} + 26148508186839885 T^{4} - $$21\!\cdots\!82$$ T^{5} + $$25\!\cdots\!43$$ T^{6} - $$16\!\cdots\!32$$ T^{7} + $$25\!\cdots\!43$$ p^{5} T^{8} - $$21\!\cdots\!82$$ p^{10} T^{9} + 26148508186839885 p^{15} T^{10} - 1670989337316 p^{20} T^{11} + 211423091 p^{25} T^{12} - 5086 p^{30} T^{13} + p^{35} T^{14} $
	41	$ 1 - 19826 T + 593046791 T^{2} - 9459032472124 T^{3} + 171616001182809933 T^{4} - $$21\!\cdots\!54$$ T^{5} + $$29\!\cdots\!91$$ T^{6} - $$31\!\cdots\!84$$ T^{7} + $$29\!\cdots\!91$$ p^{5} T^{8} - $$21\!\cdots\!54$$ p^{10} T^{9} + 171616001182809933 p^{15} T^{10} - 9459032472124 p^{20} T^{11} + 593046791 p^{25} T^{12} - 19826 p^{30} T^{13} + p^{35} T^{14} $
	43	$ 1 + 19498 T + 712545358 T^{2} + 11761185213252 T^{3} + 246401761263099659 T^{4} + $$33\!\cdots\!08$$ T^{5} + $$52\!\cdots\!09$$ T^{6} + $$61\!\cdots\!24$$ T^{7} + $$52\!\cdots\!09$$ p^{5} T^{8} + $$33\!\cdots\!08$$ p^{10} T^{9} + 246401761263099659 p^{15} T^{10} + 11761185213252 p^{20} T^{11} + 712545358 p^{25} T^{12} + 19498 p^{30} T^{13} + p^{35} T^{14} $
	47	$ 1 + 14278 T + 1201822682 T^{2} + 20857882190560 T^{3} + 651471936398667127 T^{4} + $$12\!\cdots\!12$$ T^{5} + $$21\!\cdots\!09$$ T^{6} + $$38\!\cdots\!32$$ T^{7} + $$21\!\cdots\!09$$ p^{5} T^{8} + $$12\!\cdots\!12$$ p^{10} T^{9} + 651471936398667127 p^{15} T^{10} + 20857882190560 p^{20} T^{11} + 1201822682 p^{25} T^{12} + 14278 p^{30} T^{13} + p^{35} T^{14} $
	53	$ 1 + 58644 T + 2871409310 T^{2} + 107556927268890 T^{3} + 3551296852033618759 T^{4} + $$96\!\cdots\!08$$ T^{5} + $$23\!\cdots\!35$$ T^{6} + $$51\!\cdots\!78$$ T^{7} + $$23\!\cdots\!35$$ p^{5} T^{8} + $$96\!\cdots\!08$$ p^{10} T^{9} + 3551296852033618759 p^{15} T^{10} + 107556927268890 p^{20} T^{11} + 2871409310 p^{25} T^{12} + 58644 p^{30} T^{13} + p^{35} T^{14} $
	59	$ 1 + 12888 T + 2787793113 T^{2} + 30982723319648 T^{3} + 4135250822408448625 T^{4} + $$43\!\cdots\!96$$ T^{5} + $$41\!\cdots\!53$$ T^{6} + $$37\!\cdots\!36$$ T^{7} + $$41\!\cdots\!53$$ p^{5} T^{8} + $$43\!\cdots\!96$$ p^{10} T^{9} + 4135250822408448625 p^{15} T^{10} + 30982723319648 p^{20} T^{11} + 2787793113 p^{25} T^{12} + 12888 p^{30} T^{13} + p^{35} T^{14} $
	61	$ 1 - 102866 T + 8092947255 T^{2} - 447622765339924 T^{3} + 20719153732113431721 T^{4} - $$80\!\cdots\!30$$ T^{5} + $$27\!\cdots\!87$$ T^{6} - $$84\!\cdots\!64$$ T^{7} + $$27\!\cdots\!87$$ p^{5} T^{8} - $$80\!\cdots\!30$$ p^{10} T^{9} + 20719153732113431721 p^{15} T^{10} - 447622765339924 p^{20} T^{11} + 8092947255 p^{25} T^{12} - 102866 p^{30} T^{13} + p^{35} T^{14} $
	67	$ 1 + 102996 T + 8803556997 T^{2} + 505523079909608 T^{3} + 26285502685329416765 T^{4} + $$11\!\cdots\!12$$ T^{5} + $$68\!\cdots\!07$$ p T^{6} + $$16\!\cdots\!00$$ T^{7} + $$68\!\cdots\!07$$ p^{6} T^{8} + $$11\!\cdots\!12$$ p^{10} T^{9} + 26285502685329416765 p^{15} T^{10} + 505523079909608 p^{20} T^{11} + 8803556997 p^{25} T^{12} + 102996 p^{30} T^{13} + p^{35} T^{14} $
	71	$ 1 - 51596 T + 7223844565 T^{2} - 185847471787640 T^{3} + 14568682889372676417 T^{4} + $$14\!\cdots\!16$$ T^{5} + $$28\!\cdots\!77$$ T^{6} + $$11\!\cdots\!72$$ T^{7} + $$28\!\cdots\!77$$ p^{5} T^{8} + $$14\!\cdots\!16$$ p^{10} T^{9} + 14568682889372676417 p^{15} T^{10} - 185847471787640 p^{20} T^{11} + 7223844565 p^{25} T^{12} - 51596 p^{30} T^{13} + p^{35} T^{14} $
	73	$ 1 + 17566 T + 8460394943 T^{2} + 69448804957812 T^{3} + 35649675593103833925 T^{4} + $$17\!\cdots\!78$$ T^{5} + $$10\!\cdots\!71$$ T^{6} + $$45\!\cdots\!52$$ T^{7} + $$10\!\cdots\!71$$ p^{5} T^{8} + $$17\!\cdots\!78$$ p^{10} T^{9} + 35649675593103833925 p^{15} T^{10} + 69448804957812 p^{20} T^{11} + 8460394943 p^{25} T^{12} + 17566 p^{30} T^{13} + p^{35} T^{14} $
	79	$ 1 + 212058 T + 33581514506 T^{2} + 3744925943723744 T^{3} + $$35\!\cdots\!23$$ T^{4} + $$34\!\cdots\!60$$ p T^{5} + $$18\!\cdots\!93$$ T^{6} + $$11\!\cdots\!16$$ T^{7} + $$18\!\cdots\!93$$ p^{5} T^{8} + $$34\!\cdots\!60$$ p^{11} T^{9} + $$35\!\cdots\!23$$ p^{15} T^{10} + 3744925943723744 p^{20} T^{11} + 33581514506 p^{25} T^{12} + 212058 p^{30} T^{13} + p^{35} T^{14} $
	83	$ 1 - 122928 T + 12107315585 T^{2} - 731849073992464 T^{3} + 63955868724528540161 T^{4} - $$39\!\cdots\!12$$ T^{5} + $$27\!\cdots\!49$$ T^{6} - $$13\!\cdots\!00$$ T^{7} + $$27\!\cdots\!49$$ p^{5} T^{8} - $$39\!\cdots\!12$$ p^{10} T^{9} + 63955868724528540161 p^{15} T^{10} - 731849073992464 p^{20} T^{11} + 12107315585 p^{25} T^{12} - 122928 p^{30} T^{13} + p^{35} T^{14} $
	89	$ 1 + 66510 T + 15099032951 T^{2} + 957479254901124 T^{3} + $$13\!\cdots\!77$$ T^{4} + $$10\!\cdots\!06$$ T^{5} + $$96\!\cdots\!19$$ T^{6} + $$70\!\cdots\!00$$ T^{7} + $$96\!\cdots\!19$$ p^{5} T^{8} + $$10\!\cdots\!06$$ p^{10} T^{9} + $$13\!\cdots\!77$$ p^{15} T^{10} + 957479254901124 p^{20} T^{11} + 15099032951 p^{25} T^{12} + 66510 p^{30} T^{13} + p^{35} T^{14} $
	97	$ 1 + 118182 T + 50075773455 T^{2} + 4684617842652020 T^{3} + $$11\!\cdots\!77$$ T^{4} + $$86\!\cdots\!70$$ T^{5} + $$14\!\cdots\!39$$ T^{6} + $$93\!\cdots\!24$$ T^{7} + $$14\!\cdots\!39$$ p^{5} T^{8} + $$86\!\cdots\!70$$ p^{10} T^{9} + $$11\!\cdots\!77$$ p^{15} T^{10} + 4684617842652020 p^{20} T^{11} + 50075773455 p^{25} T^{12} + 118182 p^{30} T^{13} + p^{35} T^{14} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.09206762075595970478757957714, −5.04686436234686137485724007279, −4.96340913058720736639910694676, −4.63354387697478698510248844599, −4.36260380550731123909751431345, −4.22825068638107652843352004166, −4.00362949184224139482052061733, −3.99497442748019539191717137856, −3.76746192345694361023648427297, −3.71797914252902898003666416068, −3.42174760785927041323854746571, −3.15741222159101438052906970724, −2.85079384480967471476392060261, −2.72571775175791300913469873168, −2.70902907242585623920644654995, −2.70431850327739513327141225295, −2.53092500557635356469450932820, −2.32247950444102598920501786400, −1.87917626823138400356429575476, −1.77480860416349327918223365249, −1.71125225746502583431121175513, −1.26350802714068901200933363607, −1.18838873375580979315191125233, −1.11600986005915270838000614450, −0.78706283178381512172470686841, 0, 0, 0, 0, 0, 0, 0, 0.78706283178381512172470686841, 1.11600986005915270838000614450, 1.18838873375580979315191125233, 1.26350802714068901200933363607, 1.71125225746502583431121175513, 1.77480860416349327918223365249, 1.87917626823138400356429575476, 2.32247950444102598920501786400, 2.53092500557635356469450932820, 2.70431850327739513327141225295, 2.70902907242585623920644654995, 2.72571775175791300913469873168, 2.85079384480967471476392060261, 3.15741222159101438052906970724, 3.42174760785927041323854746571, 3.71797914252902898003666416068, 3.76746192345694361023648427297, 3.99497442748019539191717137856, 4.00362949184224139482052061733, 4.22825068638107652843352004166, 4.36260380550731123909751431345, 4.63354387697478698510248844599, 4.96340913058720736639910694676, 5.04686436234686137485724007279, 5.09206762075595970478757957714

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(3)\)	\(=\)	\(0\)
\(L(\frac12)\)	\(=\)	\(0\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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