Properties

Label 4-304e2-1.1-c1e2-0-14
Degree $4$
Conductor $92416$
Sign $-1$
Analytic cond. $5.89252$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 9-s − 2·17-s − 10·23-s − 6·25-s − 12·31-s + 8·41-s + 8·47-s − 7·49-s − 2·63-s − 4·71-s + 6·73-s − 12·79-s − 8·81-s − 12·89-s + 20·97-s + 4·113-s + 4·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯
L(s)  = 1  − 0.755·7-s + 1/3·9-s − 0.485·17-s − 2.08·23-s − 6/5·25-s − 2.15·31-s + 1.24·41-s + 1.16·47-s − 49-s − 0.251·63-s − 0.474·71-s + 0.702·73-s − 1.35·79-s − 8/9·81-s − 1.27·89-s + 2.03·97-s + 0.376·113-s + 0.366·119-s + 6/11·121-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 − 0.161·153-s + 0.0798·157-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(92416\)    =    \(2^{8} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(5.89252\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 92416,\ (\ :1/2, 1/2),\ -1)\)

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$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
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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

−9.336126258147034814112503800632, −9.064491317633067178100870937304, −8.389030956186099906416038465423, −7.79042982851671684778978336755, −7.42901332013629996889715545116, −6.90558712113642696385471185618, −6.13618729618275086988924178004, −5.92431601171042959229950212980, −5.32611062091362734112131005818, −4.34996062031826374295070956274, −3.99294556253432366313577205656, −3.39849180784135208243658224296, −2.41089010965825369147834256248, −1.74314978812294076481086490013, 0, 1.74314978812294076481086490013, 2.41089010965825369147834256248, 3.39849180784135208243658224296, 3.99294556253432366313577205656, 4.34996062031826374295070956274, 5.32611062091362734112131005818, 5.92431601171042959229950212980, 6.13618729618275086988924178004, 6.90558712113642696385471185618, 7.42901332013629996889715545116, 7.79042982851671684778978336755, 8.389030956186099906416038465423, 9.064491317633067178100870937304, 9.336126258147034814112503800632

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