↑

Properties

Label 8-14e8-1.1-c9e4-0-0
Degree $8$
Conductor $1475789056$
Sign $1$
Analytic cond. $1.03842\times 10^{8}$
Root an. cond. $10.0472$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 224·3-s + 1.59e3·5-s + 3.36e4·9-s + 5.05e4·11-s + 3.69e5·13-s − 3.57e5·15-s − 1.52e5·17-s − 6.05e5·19-s + 2.50e6·23-s + 3.06e6·25-s − 2.56e6·27-s − 5.45e6·29-s − 6.65e6·31-s − 1.13e7·33-s + 6.86e6·37-s − 8.27e7·39-s − 2.83e7·41-s − 2.26e6·43-s + 5.37e7·45-s + 3.54e7·47-s + 3.40e7·51-s + 3.33e7·53-s + 8.07e7·55-s + 1.35e8·57-s + 2.94e7·59-s + 3.59e7·61-s + 5.89e8·65-s + ⋯
L(s)  = 1  − 1.59·3-s + 1.14·5-s + 1.71·9-s + 1.04·11-s + 3.58·13-s − 1.82·15-s − 0.441·17-s − 1.06·19-s + 1.86·23-s + 1.56·25-s − 0.929·27-s − 1.43·29-s − 1.29·31-s − 1.66·33-s + 0.601·37-s − 5.72·39-s − 1.56·41-s − 0.101·43-s + 1.95·45-s + 1.06·47-s + 0.705·51-s + 0.579·53-s + 1.18·55-s + 1.70·57-s + 0.316·59-s + 0.332·61-s + 4.09·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.03842\times 10^{8}\)
Root analytic conductor: \(10.0472\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 7^{8} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.2784338498\)
\(L(\frac12)\) \(\approx\) \(0.2784338498\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 224 T + 16510 T^{2} - 425600 p T^{3} - 26848349 p^{2} T^{4} - 425600 p^{10} T^{5} + 16510 p^{18} T^{6} + 224 p^{27} T^{7} + p^{36} T^{8} \)
5$D_4\times C_2$ \( 1 - 1596 T - 518074 T^{2} + 268434432 p T^{3} + 24250970739 p^{2} T^{4} + 268434432 p^{10} T^{5} - 518074 p^{18} T^{6} - 1596 p^{27} T^{7} + p^{36} T^{8} \)
11$D_4\times C_2$ \( 1 - 50592 T - 2281313878 T^{2} - 6322429624320 T^{3} + 11838764157568082859 T^{4} - 6322429624320 p^{9} T^{5} - 2281313878 p^{18} T^{6} - 50592 p^{27} T^{7} + p^{36} T^{8} \)
13$D_{4}$ \( ( 1 - 184772 T + 23500618842 T^{2} - 184772 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 152124 T - 2999635238 p T^{2} - 24802333861022928 T^{3} - \)\(11\!\cdots\!97\)\( T^{4} - 24802333861022928 p^{9} T^{5} - 2999635238 p^{19} T^{6} + 152124 p^{27} T^{7} + p^{36} T^{8} \)
19$D_4\times C_2$ \( 1 + 605024 T - 15295186070 p T^{2} + 6829015002868352 T^{3} + \)\(20\!\cdots\!91\)\( T^{4} + 6829015002868352 p^{9} T^{5} - 15295186070 p^{19} T^{6} + 605024 p^{27} T^{7} + p^{36} T^{8} \)
23$D_4\times C_2$ \( 1 - 2503608 T + 1104173268626 T^{2} - 3909570225809412096 T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - 3909570225809412096 p^{9} T^{5} + 1104173268626 p^{18} T^{6} - 2503608 p^{27} T^{7} + p^{36} T^{8} \)
29$D_{4}$ \( ( 1 + 2727492 T + 30814589817070 T^{2} + 2727492 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 6655040 T + 16022076232402 T^{2} - \)\(16\!\cdots\!60\)\( T^{3} - \)\(10\!\cdots\!37\)\( T^{4} - \)\(16\!\cdots\!60\)\( p^{9} T^{5} + 16022076232402 p^{18} T^{6} + 6655040 p^{27} T^{7} + p^{36} T^{8} \)
37$D_4\times C_2$ \( 1 - 6860228 T - 42485512358510 T^{2} + \)\(11\!\cdots\!80\)\( T^{3} - \)\(14\!\cdots\!61\)\( T^{4} + \)\(11\!\cdots\!80\)\( p^{9} T^{5} - 42485512358510 p^{18} T^{6} - 6860228 p^{27} T^{7} + p^{36} T^{8} \)
41$D_{4}$ \( ( 1 + 14154756 T + 639842959527622 T^{2} + 14154756 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 1133840 T + 758217452534742 T^{2} + 1133840 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 35486640 T - 1208713365428110 T^{2} - \)\(81\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!11\)\( T^{4} - \)\(81\!\cdots\!40\)\( p^{9} T^{5} - 1208713365428110 p^{18} T^{6} - 35486640 p^{27} T^{7} + p^{36} T^{8} \)
53$D_4\times C_2$ \( 1 - 33311940 T - 1444260378991966 T^{2} + \)\(13\!\cdots\!00\)\( T^{3} - \)\(80\!\cdots\!33\)\( T^{4} + \)\(13\!\cdots\!00\)\( p^{9} T^{5} - 1444260378991966 p^{18} T^{6} - 33311940 p^{27} T^{7} + p^{36} T^{8} \)
59$D_4\times C_2$ \( 1 - 29419488 T - 3232851982013266 T^{2} + \)\(38\!\cdots\!84\)\( T^{3} - \)\(65\!\cdots\!37\)\( T^{4} + \)\(38\!\cdots\!84\)\( p^{9} T^{5} - 3232851982013266 p^{18} T^{6} - 29419488 p^{27} T^{7} + p^{36} T^{8} \)
61$D_4\times C_2$ \( 1 - 589372 p T - 16341552292782410 T^{2} + \)\(33\!\cdots\!76\)\( p T^{3} + \)\(15\!\cdots\!71\)\( T^{4} + \)\(33\!\cdots\!76\)\( p^{10} T^{5} - 16341552292782410 p^{18} T^{6} - 589372 p^{28} T^{7} + p^{36} T^{8} \)
67$D_4\times C_2$ \( 1 - 170286704 T - 27171545065454582 T^{2} - \)\(29\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!16\)\( p^{9} T^{5} - 27171545065454582 p^{18} T^{6} - 170286704 p^{27} T^{7} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 + 463615488 T + 145415881494979342 T^{2} + 463615488 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 100806860 T - 37231158961574630 T^{2} - \)\(70\!\cdots\!60\)\( T^{3} - \)\(18\!\cdots\!69\)\( T^{4} - \)\(70\!\cdots\!60\)\( p^{9} T^{5} - 37231158961574630 p^{18} T^{6} + 100806860 p^{27} T^{7} + p^{36} T^{8} \)
79$D_4\times C_2$ \( 1 - 197493824 T - 191512573230546782 T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!20\)\( p^{9} T^{5} - 191512573230546782 p^{18} T^{6} - 197493824 p^{27} T^{7} + p^{36} T^{8} \)
83$D_{4}$ \( ( 1 + 51427488 T + 307270000647008386 T^{2} + 51427488 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 649287492 T - 341509972166851174 T^{2} - \)\(40\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!40\)\( p^{9} T^{5} - 341509972166851174 p^{18} T^{6} - 649287492 p^{27} T^{7} + p^{36} T^{8} \)
97$D_{4}$ \( ( 1 + 2781293620 T + 3412420553054923350 T^{2} + 2781293620 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.21288601455843923383870809746, −7.17347852760292658437265092272, −6.80301237817410429884032263992, −6.46021320258447235169273675659, −6.22934748836368084695844262968, −6.05348119845853497623302676328, −5.99365212897210611191562919946, −5.63179018633053186647037822770, −5.23137299349518649807631515851, −5.03032101816907154750759249354, −4.92642851237049299019651423231, −4.20856829130029921321797049101, −3.99439525781661635149747540784, −3.91046225120070487159002192196, −3.69779266353331047777595719221, −3.13031836658044444730239564295, −2.83493878050780295150886571192, −2.46536166904117347804289306357, −2.01240158240022863122412819789, −1.38904913712174620731589216065, −1.36729355564084943373080210381, −1.34677460558024951426108735865, −1.10580402805226663280400554854, −0.54879930211477843169362863672, −0.05878491079205873626724503119, 0.05878491079205873626724503119, 0.54879930211477843169362863672, 1.10580402805226663280400554854, 1.34677460558024951426108735865, 1.36729355564084943373080210381, 1.38904913712174620731589216065, 2.01240158240022863122412819789, 2.46536166904117347804289306357, 2.83493878050780295150886571192, 3.13031836658044444730239564295, 3.69779266353331047777595719221, 3.91046225120070487159002192196, 3.99439525781661635149747540784, 4.20856829130029921321797049101, 4.92642851237049299019651423231, 5.03032101816907154750759249354, 5.23137299349518649807631515851, 5.63179018633053186647037822770, 5.99365212897210611191562919946, 6.05348119845853497623302676328, 6.22934748836368084695844262968, 6.46021320258447235169273675659, 6.80301237817410429884032263992, 7.17347852760292658437265092272, 7.21288601455843923383870809746

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