Properties

Label 4-304e2-1.1-c4e2-0-0
Degree $4$
Conductor $92416$
Sign $1$
Analytic cond. $987.497$
Root an. cond. $5.60575$
Motivic weight $4$
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  − 31·5-s − 73·7-s + 162·9-s − 233·11-s + 353·17-s − 722·19-s + 316·23-s + 625·25-s + 2.26e3·35-s + 3.52e3·43-s − 5.02e3·45-s + 1.20e3·47-s + 2.40e3·49-s + 7.22e3·55-s − 3.16e3·61-s − 1.18e4·63-s + 1.00e4·73-s + 1.70e4·77-s + 1.96e4·81-s + 1.13e4·83-s − 1.09e4·85-s + 2.23e4·95-s − 3.77e4·99-s − 1.99e4·101-s − 9.79e3·115-s − 2.57e4·119-s + 1.46e4·121-s + ⋯
L(s)  = 1  − 1.23·5-s − 1.48·7-s + 2·9-s − 1.92·11-s + 1.22·17-s − 2·19-s + 0.597·23-s + 25-s + 1.84·35-s + 1.90·43-s − 2.47·45-s + 0.546·47-s + 49-s + 2.38·55-s − 0.851·61-s − 2.97·63-s + 1.88·73-s + 2.86·77-s + 3·81-s + 1.64·83-s − 1.51·85-s + 2.47·95-s − 3.85·99-s − 1.96·101-s − 0.740·115-s − 1.81·119-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(987.497\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 92416,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8426581739\)
\(L(\frac12)\) \(\approx\) \(0.8426581739\)
\(L(3)\) 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 \( 1 \)
19$C_1$ \( ( 1 + p^{2} T )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
5$C_2^2$ \( 1 + 31 T + 336 T^{2} + 31 p^{4} T^{3} + p^{8} T^{4} \)
7$C_2^2$ \( 1 + 73 T + 2928 T^{2} + 73 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2^2$ \( 1 + 233 T + 39648 T^{2} + 233 p^{4} T^{3} + p^{8} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
17$C_2^2$ \( 1 - 353 T + 41088 T^{2} - 353 p^{4} T^{3} + p^{8} T^{4} \)
23$C_2$ \( ( 1 - 158 T + p^{4} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
43$C_2^2$ \( 1 - 3527 T + 9020928 T^{2} - 3527 p^{4} T^{3} + p^{8} T^{4} \)
47$C_2^2$ \( 1 - 1207 T - 3422832 T^{2} - 1207 p^{4} T^{3} + p^{8} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
61$C_2^2$ \( 1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
73$C_2^2$ \( 1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
83$C_2$ \( ( 1 - 5678 T + p^{4} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
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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

−10.95240219815696386526109384201, −10.85892984827640419235482496352, −10.32115537757384892687341971643, −10.14013856401201410031513731708, −9.392200407310255401152114940190, −9.158601694087756824725542624961, −8.107200147727656541209355193167, −8.065892195228873796997326352875, −7.29464165242088584373202853556, −7.22064065071941939568119767965, −6.46842371030224318748053220339, −6.03376298507609306227627918336, −5.12493154858281719177787667525, −4.70829232359955676675288697186, −3.86928592816579607662981692405, −3.78434703301554825974946216427, −2.83370906936667219368000080827, −2.30626994982906196938566675583, −1.08735296575340697381193624608, −0.32603285767085400083839343168, 0.32603285767085400083839343168, 1.08735296575340697381193624608, 2.30626994982906196938566675583, 2.83370906936667219368000080827, 3.78434703301554825974946216427, 3.86928592816579607662981692405, 4.70829232359955676675288697186, 5.12493154858281719177787667525, 6.03376298507609306227627918336, 6.46842371030224318748053220339, 7.22064065071941939568119767965, 7.29464165242088584373202853556, 8.065892195228873796997326352875, 8.107200147727656541209355193167, 9.158601694087756824725542624961, 9.392200407310255401152114940190, 10.14013856401201410031513731708, 10.32115537757384892687341971643, 10.85892984827640419235482496352, 10.95240219815696386526109384201

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