Properties

Label 4-882e2-1.1-c3e2-0-3
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
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  + 2·2-s − 6·5-s − 8·8-s − 12·10-s + 30·11-s − 4·13-s − 16·16-s − 66·17-s − 52·19-s + 60·22-s + 114·23-s + 125·25-s − 8·26-s − 144·29-s − 196·31-s − 132·34-s + 286·37-s − 104·38-s + 48·40-s − 756·41-s + 328·43-s + 228·46-s + 228·47-s + 250·50-s − 348·53-s − 180·55-s − 288·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.536·5-s − 0.353·8-s − 0.379·10-s + 0.822·11-s − 0.0853·13-s − 1/4·16-s − 0.941·17-s − 0.627·19-s + 0.581·22-s + 1.03·23-s + 25-s − 0.0603·26-s − 0.922·29-s − 1.13·31-s − 0.665·34-s + 1.27·37-s − 0.443·38-s + 0.189·40-s − 2.87·41-s + 1.16·43-s + 0.730·46-s + 0.707·47-s + 0.707·50-s − 0.901·53-s − 0.441·55-s − 0.652·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8321770494\)
\(L(\frac12)\) \(\approx\) \(0.8321770494\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 6 T - 89 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 30 T - 431 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 66 T - 557 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 52 T - 4155 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 114 T + 829 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 72 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 196 T + 8625 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 286 T + 31143 T^{2} - 286 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 378 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 228 T - 51839 T^{2} - 228 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 348 T - 27773 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 348 T - 84275 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 106 T - 215745 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 596 T + 54453 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 630 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1042 T + 696747 T^{2} + 1042 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 88 T - 485295 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 1440 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 1374 T + 1182907 T^{2} + 1374 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2} \)
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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

−10.06946770712219123274139885853, −9.467618317287535754497551797318, −8.861101604941372303105554871631, −8.755853102314902768029984070298, −8.603458514274713797807768888849, −7.52703542281427559816487036403, −7.49537530553783123397942145595, −6.83186050129593487593997467383, −6.64804965104437394035672070558, −5.82817501837948010021124025733, −5.75976421998464897104921512349, −4.85242060712740558978671319614, −4.68981066610943421652523364459, −4.00390579259311710752307069156, −3.87562107281846589953842992237, −2.96320253420233940635106291917, −2.79364914463039710885903975291, −1.76244568606205722325927727087, −1.30001616275690227137719217520, −0.21270023740601513564615700859, 0.21270023740601513564615700859, 1.30001616275690227137719217520, 1.76244568606205722325927727087, 2.79364914463039710885903975291, 2.96320253420233940635106291917, 3.87562107281846589953842992237, 4.00390579259311710752307069156, 4.68981066610943421652523364459, 4.85242060712740558978671319614, 5.75976421998464897104921512349, 5.82817501837948010021124025733, 6.64804965104437394035672070558, 6.83186050129593487593997467383, 7.49537530553783123397942145595, 7.52703542281427559816487036403, 8.603458514274713797807768888849, 8.755853102314902768029984070298, 8.861101604941372303105554871631, 9.467618317287535754497551797318, 10.06946770712219123274139885853

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