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Properties

Label 8-390e4-1.1-c5e4-0-0
Degree $8$
Conductor $23134410000$
Sign $1$
Analytic cond. $1.53073\times 10^{7}$
Root an. cond. $7.90883$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 36·3-s + 160·4-s + 100·5-s + 576·6-s + 247·7-s + 1.28e3·8-s + 810·9-s + 1.60e3·10-s + 723·11-s + 5.76e3·12-s − 676·13-s + 3.95e3·14-s + 3.60e3·15-s + 8.96e3·16-s + 1.66e3·17-s + 1.29e4·18-s + 2.66e3·19-s + 1.60e4·20-s + 8.89e3·21-s + 1.15e4·22-s + 4.53e3·23-s + 4.60e4·24-s + 6.25e3·25-s − 1.08e4·26-s + 1.45e4·27-s + 3.95e4·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.90·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s + 1.80·11-s + 11.5·12-s − 1.10·13-s + 5.38·14-s + 4.13·15-s + 35/4·16-s + 1.40·17-s + 9.42·18-s + 1.69·19-s + 8.94·20-s + 4.39·21-s + 5.09·22-s + 1.78·23-s + 16.3·24-s + 2·25-s − 3.13·26-s + 3.84·27-s + 9.52·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.53073\times 10^{7}\)
Root analytic conductor: \(7.90883\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(778.1288468\)
\(L(\frac12)\) \(\approx\) \(778.1288468\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{4} \)
3$C_1$ \( ( 1 - p^{2} T )^{4} \)
5$C_1$ \( ( 1 - p^{2} T )^{4} \)
13$C_1$ \( ( 1 + p^{2} T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 247 T + 48302 T^{2} - 7639411 T^{3} + 1054098674 T^{4} - 7639411 p^{5} T^{5} + 48302 p^{10} T^{6} - 247 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 723 T + 527034 T^{2} - 269511123 T^{3} + 121509557962 T^{4} - 269511123 p^{5} T^{5} + 527034 p^{10} T^{6} - 723 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 1669 T + 5695130 T^{2} - 6882749923 T^{3} + 715614753594 p T^{4} - 6882749923 p^{5} T^{5} + 5695130 p^{10} T^{6} - 1669 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 2668 T + 3468560 T^{2} - 4524049852 T^{3} + 7179219186542 T^{4} - 4524049852 p^{5} T^{5} + 3468560 p^{10} T^{6} - 2668 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 4535 T + 11033340 T^{2} - 25351980555 T^{3} + 81272631583942 T^{4} - 25351980555 p^{5} T^{5} + 11033340 p^{10} T^{6} - 4535 p^{15} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 8888 T + 75559348 T^{2} - 318614918888 T^{3} + 1762049217224934 T^{4} - 318614918888 p^{5} T^{5} + 75559348 p^{10} T^{6} - 8888 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 588 T + 46682800 T^{2} + 332050168316 T^{3} + 702639348188766 T^{4} + 332050168316 p^{5} T^{5} + 46682800 p^{10} T^{6} + 588 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 11191 T + 125893298 T^{2} + 71965332349 T^{3} + 1523309146853402 T^{4} + 71965332349 p^{5} T^{5} + 125893298 p^{10} T^{6} + 11191 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 10905 T - 119884050 T^{2} - 196854682703 T^{3} + 34197178252709698 T^{4} - 196854682703 p^{5} T^{5} - 119884050 p^{10} T^{6} - 10905 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 2348 T + 329111740 T^{2} - 612603152596 T^{3} + 53053735046770038 T^{4} - 612603152596 p^{5} T^{5} + 329111740 p^{10} T^{6} + 2348 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 9148 T + 287215392 T^{2} - 1327849488180 T^{3} + 23616815602566526 T^{4} - 1327849488180 p^{5} T^{5} + 287215392 p^{10} T^{6} + 9148 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 13449 T + 1073171526 T^{2} - 18691511363299 T^{3} + 558520984945542898 T^{4} - 18691511363299 p^{5} T^{5} + 1073171526 p^{10} T^{6} - 13449 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 25358 T + 2738843164 T^{2} - 49547394476814 T^{3} + 2887717654605091190 T^{4} - 49547394476814 p^{5} T^{5} + 2738843164 p^{10} T^{6} - 25358 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 53411 T + 3581546506 T^{2} - 125997666139833 T^{3} + 4658221298349706218 T^{4} - 125997666139833 p^{5} T^{5} + 3581546506 p^{10} T^{6} - 53411 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 630 T + 779989300 T^{2} - 26489858628870 T^{3} - 1342726112188280986 T^{4} - 26489858628870 p^{5} T^{5} + 779989300 p^{10} T^{6} - 630 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 23337 T + 2319517574 T^{2} - 40654852164243 T^{3} + 1217966928667662002 T^{4} - 40654852164243 p^{5} T^{5} + 2319517574 p^{10} T^{6} + 23337 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 38236 T + 1692664108 T^{2} - 123773089971108 T^{3} + 9546100866252551046 T^{4} - 123773089971108 p^{5} T^{5} + 1692664108 p^{10} T^{6} - 38236 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 32029 T + 4005128224 T^{2} - 130088796436583 T^{3} + 3267271207045380222 T^{4} - 130088796436583 p^{5} T^{5} + 4005128224 p^{10} T^{6} + 32029 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 7986 T + 5263232660 T^{2} - 290282673141090 T^{3} + 9502942627836010214 T^{4} - 290282673141090 p^{5} T^{5} + 5263232660 p^{10} T^{6} - 7986 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 148095 T + 11920908378 T^{2} - 23516340614665 T^{3} - 23762925532771709846 T^{4} - 23516340614665 p^{5} T^{5} + 11920908378 p^{10} T^{6} - 148095 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 63455 T + 32452708442 T^{2} - 1494238837974569 T^{3} + \)\(40\!\cdots\!70\)\( T^{4} - 1494238837974569 p^{5} T^{5} + 32452708442 p^{10} T^{6} - 63455 p^{15} T^{7} + p^{20} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.45355401960790007066762949940, −6.71615713692983705142839899796, −6.71147603609304395040079333521, −6.68063214988531961637448058573, −6.62535518623678070459071370100, −5.75634202922705312142480664099, −5.56791230657526876488817509976, −5.49964790550963579252211937745, −5.28568419211144904606460316653, −4.80465177263257118463941132024, −4.68821465629062993209063718123, −4.47826173036827799702451234564, −4.45172525789179533204732114244, −3.61414831597269000703295974811, −3.46841276209944649480064124204, −3.37976793617020632542544475314, −3.27739717182681600580090788247, −2.51937353152128998612304662511, −2.51032711604385499474742545637, −2.17599464923063295339905458700, −2.14841145557093472780548670794, −1.32954425272739189514354674792, −1.32010395988436179203332493465, −1.09483254545776620401841624817, −1.04897107746182494064852900107, 1.04897107746182494064852900107, 1.09483254545776620401841624817, 1.32010395988436179203332493465, 1.32954425272739189514354674792, 2.14841145557093472780548670794, 2.17599464923063295339905458700, 2.51032711604385499474742545637, 2.51937353152128998612304662511, 3.27739717182681600580090788247, 3.37976793617020632542544475314, 3.46841276209944649480064124204, 3.61414831597269000703295974811, 4.45172525789179533204732114244, 4.47826173036827799702451234564, 4.68821465629062993209063718123, 4.80465177263257118463941132024, 5.28568419211144904606460316653, 5.49964790550963579252211937745, 5.56791230657526876488817509976, 5.75634202922705312142480664099, 6.62535518623678070459071370100, 6.68063214988531961637448058573, 6.71147603609304395040079333521, 6.71615713692983705142839899796, 7.45355401960790007066762949940

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