Properties

Label 10-69e5-1.1-c7e5-0-0
Degree $10$
Conductor $1564031349$
Sign $-1$
Analytic cond. $4.65260\times 10^{6}$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 135·3-s − 185·4-s − 266·5-s − 496·7-s + 474·8-s + 1.09e4·9-s − 1.14e3·11-s + 2.49e4·12-s − 642·13-s + 3.59e4·15-s + 2.82e3·16-s − 5.79e3·17-s − 6.03e3·19-s + 4.92e4·20-s + 6.69e4·21-s + 6.08e4·23-s − 6.39e4·24-s − 2.91e5·25-s − 6.88e5·27-s + 9.17e4·28-s − 1.69e5·29-s − 1.99e5·31-s − 1.81e5·32-s + 1.54e5·33-s + 1.31e5·35-s − 2.02e6·36-s − 2.02e5·37-s + ⋯
L(s)  = 1  − 2.88·3-s − 1.44·4-s − 0.951·5-s − 0.546·7-s + 0.327·8-s + 5·9-s − 0.260·11-s + 4.17·12-s − 0.0810·13-s + 2.74·15-s + 0.172·16-s − 0.286·17-s − 0.201·19-s + 1.37·20-s + 1.57·21-s + 1.04·23-s − 0.944·24-s − 3.72·25-s − 6.73·27-s + 0.789·28-s − 1.28·29-s − 1.20·31-s − 0.976·32-s + 0.750·33-s + 0.520·35-s − 7.22·36-s − 0.655·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 23^{5}\)
Sign: $-1$
Analytic conductor: \(4.65260\times 10^{6}\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{69} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 23^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ -1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{3} T )^{5} \)
23$C_1$ \( ( 1 - p^{3} T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + 185 T^{2} - 237 p T^{3} + 7851 p^{2} T^{4} + 709 p^{3} T^{5} + 7851 p^{9} T^{6} - 237 p^{15} T^{7} + 185 p^{21} T^{8} + p^{35} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 266 T + 361929 T^{2} + 15472648 p T^{3} + 2161955522 p^{2} T^{4} + 14139712924 p^{4} T^{5} + 2161955522 p^{9} T^{6} + 15472648 p^{15} T^{7} + 361929 p^{21} T^{8} + 266 p^{28} T^{9} + p^{35} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 496 T + 1756571 T^{2} + 1123810704 T^{3} + 2372248497634 T^{4} + 984010614791616 T^{5} + 2372248497634 p^{7} T^{6} + 1123810704 p^{14} T^{7} + 1756571 p^{21} T^{8} + 496 p^{28} T^{9} + p^{35} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 1148 T + 79535271 T^{2} + 81036609776 T^{3} + 2759193912612338 T^{4} + 2286597617642676456 T^{5} + 2759193912612338 p^{7} T^{6} + 81036609776 p^{14} T^{7} + 79535271 p^{21} T^{8} + 1148 p^{28} T^{9} + p^{35} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 642 T + 107144993 T^{2} + 115024854072 T^{3} + 9280068844882210 T^{4} + 26329651650478741356 T^{5} + 9280068844882210 p^{7} T^{6} + 115024854072 p^{14} T^{7} + 107144993 p^{21} T^{8} + 642 p^{28} T^{9} + p^{35} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 5798 T + 765741381 T^{2} - 1663501787896 T^{3} + 370680130216694714 T^{4} - \)\(21\!\cdots\!20\)\( T^{5} + 370680130216694714 p^{7} T^{6} - 1663501787896 p^{14} T^{7} + 765741381 p^{21} T^{8} + 5798 p^{28} T^{9} + p^{35} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 6036 T + 3492386023 T^{2} + 25456470564896 T^{3} + 5555373680250650218 T^{4} + \)\(34\!\cdots\!40\)\( T^{5} + 5555373680250650218 p^{7} T^{6} + 25456470564896 p^{14} T^{7} + 3492386023 p^{21} T^{8} + 6036 p^{28} T^{9} + p^{35} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 169162 T + 80325985009 T^{2} + 353217236227768 p T^{3} + \)\(26\!\cdots\!06\)\( T^{4} + \)\(25\!\cdots\!40\)\( T^{5} + \)\(26\!\cdots\!06\)\( p^{7} T^{6} + 353217236227768 p^{15} T^{7} + 80325985009 p^{21} T^{8} + 169162 p^{28} T^{9} + p^{35} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 6440 p T + 3018327269 p T^{2} + 8543888532695840 T^{3} + \)\(29\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!72\)\( T^{5} + \)\(29\!\cdots\!94\)\( p^{7} T^{6} + 8543888532695840 p^{14} T^{7} + 3018327269 p^{22} T^{8} + 6440 p^{29} T^{9} + p^{35} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 202002 T + 369504819393 T^{2} + 64438923459378696 T^{3} + \)\(60\!\cdots\!22\)\( T^{4} + \)\(86\!\cdots\!00\)\( T^{5} + \)\(60\!\cdots\!22\)\( p^{7} T^{6} + 64438923459378696 p^{14} T^{7} + 369504819393 p^{21} T^{8} + 202002 p^{28} T^{9} + p^{35} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 322 p^{2} T + 870326731909 T^{2} - 355050034300082968 T^{3} + \)\(31\!\cdots\!30\)\( T^{4} - \)\(97\!\cdots\!60\)\( T^{5} + \)\(31\!\cdots\!30\)\( p^{7} T^{6} - 355050034300082968 p^{14} T^{7} + 870326731909 p^{21} T^{8} - 322 p^{30} T^{9} + p^{35} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 909596 T + 638610119087 T^{2} + 352087668310526336 T^{3} + \)\(24\!\cdots\!94\)\( T^{4} + \)\(28\!\cdots\!72\)\( p T^{5} + \)\(24\!\cdots\!94\)\( p^{7} T^{6} + 352087668310526336 p^{14} T^{7} + 638610119087 p^{21} T^{8} + 909596 p^{28} T^{9} + p^{35} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 80208 T + 1140209963019 T^{2} - 19449762264067776 T^{3} + \)\(89\!\cdots\!06\)\( T^{4} - \)\(25\!\cdots\!72\)\( T^{5} + \)\(89\!\cdots\!06\)\( p^{7} T^{6} - 19449762264067776 p^{14} T^{7} + 1140209963019 p^{21} T^{8} - 80208 p^{28} T^{9} + p^{35} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 278138 T + 2816263082553 T^{2} + 807494496605031560 T^{3} + \)\(38\!\cdots\!78\)\( T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(38\!\cdots\!78\)\( p^{7} T^{6} + 807494496605031560 p^{14} T^{7} + 2816263082553 p^{21} T^{8} + 278138 p^{28} T^{9} + p^{35} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 3177380 T + 9136544467935 T^{2} + 17964595219447971248 T^{3} + \)\(37\!\cdots\!66\)\( T^{4} + \)\(57\!\cdots\!28\)\( T^{5} + \)\(37\!\cdots\!66\)\( p^{7} T^{6} + 17964595219447971248 p^{14} T^{7} + 9136544467935 p^{21} T^{8} + 3177380 p^{28} T^{9} + p^{35} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 147782 T + 5932620594121 T^{2} - 9675976261596442392 T^{3} + \)\(88\!\cdots\!26\)\( T^{4} - \)\(58\!\cdots\!24\)\( T^{5} + \)\(88\!\cdots\!26\)\( p^{7} T^{6} - 9675976261596442392 p^{14} T^{7} + 5932620594121 p^{21} T^{8} - 147782 p^{28} T^{9} + p^{35} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 464916 T + 3276270412215 T^{2} + 10979955165261254592 T^{3} + \)\(60\!\cdots\!22\)\( T^{4} + \)\(18\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!22\)\( p^{7} T^{6} + 10979955165261254592 p^{14} T^{7} + 3276270412215 p^{21} T^{8} + 464916 p^{28} T^{9} + p^{35} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 1576792 T + 41978900215987 T^{2} - 57544448125800410272 T^{3} + \)\(73\!\cdots\!94\)\( T^{4} - \)\(78\!\cdots\!48\)\( T^{5} + \)\(73\!\cdots\!94\)\( p^{7} T^{6} - 57544448125800410272 p^{14} T^{7} + 41978900215987 p^{21} T^{8} - 1576792 p^{28} T^{9} + p^{35} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 38190 T + 47748528497349 T^{2} + 644610540830176872 T^{3} + \)\(97\!\cdots\!78\)\( T^{4} + \)\(61\!\cdots\!12\)\( T^{5} + \)\(97\!\cdots\!78\)\( p^{7} T^{6} + 644610540830176872 p^{14} T^{7} + 47748528497349 p^{21} T^{8} + 38190 p^{28} T^{9} + p^{35} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 3913336 T + 76011562921523 T^{2} + \)\(29\!\cdots\!68\)\( T^{3} + \)\(25\!\cdots\!54\)\( T^{4} + \)\(83\!\cdots\!72\)\( T^{5} + \)\(25\!\cdots\!54\)\( p^{7} T^{6} + \)\(29\!\cdots\!68\)\( p^{14} T^{7} + 76011562921523 p^{21} T^{8} + 3913336 p^{28} T^{9} + p^{35} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 15774716 T + 194216440461567 T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!42\)\( T^{4} - \)\(59\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!42\)\( p^{7} T^{6} - \)\(15\!\cdots\!20\)\( p^{14} T^{7} + 194216440461567 p^{21} T^{8} - 15774716 p^{28} T^{9} + p^{35} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 1116482 T + 104114520441501 T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(49\!\cdots\!14\)\( T^{4} + \)\(19\!\cdots\!44\)\( T^{5} + \)\(49\!\cdots\!14\)\( p^{7} T^{6} + \)\(20\!\cdots\!52\)\( p^{14} T^{7} + 104114520441501 p^{21} T^{8} - 1116482 p^{28} T^{9} + p^{35} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 15738566 T + 381721971777741 T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(60\!\cdots\!82\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(60\!\cdots\!82\)\( p^{7} T^{6} + \)\(44\!\cdots\!44\)\( p^{14} T^{7} + 381721971777741 p^{21} T^{8} + 15738566 p^{28} T^{9} + p^{35} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.276826841433842585233499034133, −8.152060366775460414518200455858, −8.017565846059859975597720437858, −7.79615719197544573354869007140, −7.39637002539725286301113543080, −7.17458608379544543255410979668, −6.96322446520949471938927925207, −6.53968313003047788688451267022, −6.30497728263180800281713791594, −6.22486204712772757264146713695, −5.80495079534187389541126855548, −5.31078086548366039912166637434, −5.29004798107548688080606984433, −5.12402218594518555340243433072, −4.93948851010054585191546588326, −4.32097977971119935772375298156, −3.98640432522017386256303090141, −3.97028470065954718883757183443, −3.89783279373140133343929921176, −3.38695247592134488151006307058, −2.66026415421870153907408770417, −2.24084829327200644471333175086, −1.59959296699392714695741523845, −1.31703756846590892476662074952, −1.24661487730044404514098620275, 0, 0, 0, 0, 0, 1.24661487730044404514098620275, 1.31703756846590892476662074952, 1.59959296699392714695741523845, 2.24084829327200644471333175086, 2.66026415421870153907408770417, 3.38695247592134488151006307058, 3.89783279373140133343929921176, 3.97028470065954718883757183443, 3.98640432522017386256303090141, 4.32097977971119935772375298156, 4.93948851010054585191546588326, 5.12402218594518555340243433072, 5.29004798107548688080606984433, 5.31078086548366039912166637434, 5.80495079534187389541126855548, 6.22486204712772757264146713695, 6.30497728263180800281713791594, 6.53968313003047788688451267022, 6.96322446520949471938927925207, 7.17458608379544543255410979668, 7.39637002539725286301113543080, 7.79615719197544573354869007140, 8.017565846059859975597720437858, 8.152060366775460414518200455858, 8.276826841433842585233499034133

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