Properties

Label 4-1620e2-1.1-c3e2-0-4
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $9136.12$
Root an. cond. $9.77666$
Motivic weight $3$
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  − 5·5-s − 17·7-s + 30·11-s + 61·13-s − 240·17-s − 86·19-s − 90·23-s + 90·29-s − 8·31-s + 85·35-s + 634·37-s + 30·41-s + 220·43-s + 180·47-s + 343·49-s − 1.26e3·53-s − 150·55-s + 840·59-s − 599·61-s − 305·65-s − 107·67-s − 420·71-s − 842·73-s − 510·77-s − 353·79-s + 1.35e3·83-s + 1.20e3·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.917·7-s + 0.822·11-s + 1.30·13-s − 3.42·17-s − 1.03·19-s − 0.815·23-s + 0.576·29-s − 0.0463·31-s + 0.410·35-s + 2.81·37-s + 0.114·41-s + 0.780·43-s + 0.558·47-s + 49-s − 3.26·53-s − 0.367·55-s + 1.85·59-s − 1.25·61-s − 0.582·65-s − 0.195·67-s − 0.702·71-s − 1.34·73-s − 0.754·77-s − 0.502·79-s + 1.78·83-s + 1.53·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(9136.12\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.039226128\)
\(L(\frac12)\) \(\approx\) \(2.039226128\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + p T + p^{2} T^{2} \)
good7$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \)
11$C_2^2$ \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 61 T + 1524 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 120 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 + 43 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 90 T - 4067 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 90 T - 16289 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 + 8 T - 29727 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 317 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T - 68021 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 220 T - 31107 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 180 T - 71423 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 630 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 840 T + 500221 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 599 T + 131820 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 107 T - 289314 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 210 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 421 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 353 T - 368430 T^{2} + 353 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 1350 T + 1250713 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1020 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 997 T + 81336 T^{2} - 997 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.226179083429175188659459723174, −8.758564152916730361485598514849, −8.583972458265926972867258102255, −8.203880469738023489568000674153, −7.52676892481923726933875965066, −7.28079275959259560500989343637, −6.63185884672655922927580328079, −6.21004499864522116792549263053, −6.16144763114791683265952609151, −6.07135376585900751757701606263, −4.81819329265286837976683784383, −4.54593645428264915969114211424, −4.08329788951541362163264855892, −4.03619744595898497975071600206, −3.21914617192825412487759132227, −2.78547446229487855110917349061, −2.05761745624867851164261772657, −1.85316408164599498483002928960, −0.72535224945872648647866593279, −0.44042020706527729357166089941, 0.44042020706527729357166089941, 0.72535224945872648647866593279, 1.85316408164599498483002928960, 2.05761745624867851164261772657, 2.78547446229487855110917349061, 3.21914617192825412487759132227, 4.03619744595898497975071600206, 4.08329788951541362163264855892, 4.54593645428264915969114211424, 4.81819329265286837976683784383, 6.07135376585900751757701606263, 6.16144763114791683265952609151, 6.21004499864522116792549263053, 6.63185884672655922927580328079, 7.28079275959259560500989343637, 7.52676892481923726933875965066, 8.203880469738023489568000674153, 8.583972458265926972867258102255, 8.758564152916730361485598514849, 9.226179083429175188659459723174

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