Properties

Label 4-1216e2-1.1-c1e2-0-27
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 3·9-s − 10·17-s + 6·23-s − 6·25-s − 20·31-s − 12·41-s − 16·47-s − 11·49-s + 6·63-s + 24·71-s + 22·73-s − 32·79-s − 8·89-s − 24·97-s + 4·103-s + 16·113-s + 20·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s − 9-s − 2.42·17-s + 1.25·23-s − 6/5·25-s − 3.59·31-s − 1.87·41-s − 2.33·47-s − 1.57·49-s + 0.755·63-s + 2.84·71-s + 2.57·73-s − 3.60·79-s − 0.847·89-s − 2.43·97-s + 0.394·103-s + 1.50·113-s + 1.83·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−9.560423801687851858035516854593, −9.078914580800521084834807858492, −8.789738150830262684436493570506, −8.364989284097332836323452357892, −8.054932625377401236969486030192, −7.37759537552068899213166869092, −6.81893578599937410538296245717, −6.75490869545100678888110266108, −6.33333972766039216673932136650, −5.64627389990694439814382002943, −5.31229610751105295105052134212, −4.94020677998479479840763424810, −4.28806962137891571388993652914, −3.68374437429460524000993398476, −3.34014982079073240885196077070, −2.83619747990884525794359393850, −1.97657079152034106338366274028, −1.77849389118508217197657670653, 0, 0, 1.77849389118508217197657670653, 1.97657079152034106338366274028, 2.83619747990884525794359393850, 3.34014982079073240885196077070, 3.68374437429460524000993398476, 4.28806962137891571388993652914, 4.94020677998479479840763424810, 5.31229610751105295105052134212, 5.64627389990694439814382002943, 6.33333972766039216673932136650, 6.75490869545100678888110266108, 6.81893578599937410538296245717, 7.37759537552068899213166869092, 8.054932625377401236969486030192, 8.364989284097332836323452357892, 8.789738150830262684436493570506, 9.078914580800521084834807858492, 9.560423801687851858035516854593

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