Properties

Label 4-164e2-1.1-c1e2-0-0
Degree $4$
Conductor $26896$
Sign $-1$
Analytic cond. $1.71491$
Root an. cond. $1.14435$
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-s − 4-s − 4·5-s + 3·8-s + 2·9-s + 4·10-s − 16-s − 2·18-s + 4·20-s + 2·25-s − 5·32-s − 2·36-s − 4·37-s − 12·40-s − 6·41-s − 8·45-s − 6·49-s − 2·50-s − 4·61-s + 7·64-s + 6·72-s − 28·73-s + 4·74-s + 4·80-s − 5·81-s + 6·82-s + 8·90-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 2/3·9-s + 1.26·10-s − 1/4·16-s − 0.471·18-s + 0.894·20-s + 2/5·25-s − 0.883·32-s − 1/3·36-s − 0.657·37-s − 1.89·40-s − 0.937·41-s − 1.19·45-s − 6/7·49-s − 0.282·50-s − 0.512·61-s + 7/8·64-s + 0.707·72-s − 3.27·73-s + 0.464·74-s + 0.447·80-s − 5/9·81-s + 0.662·82-s + 0.843·90-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26896\)    =    \(2^{4} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(1.71491\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 26896,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
41$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−10.19023449659800280025304267480, −9.950081302442658861535765863833, −9.136661245630303130527527106950, −8.751287964350617633590160637874, −8.062173761857703697875994376929, −7.85645423889849417776793402919, −7.25560493967332643536479769551, −6.83969790329135228911236082140, −5.83159640611586890531957946716, −4.97376859736062822975389284623, −4.33835034948233865658543563854, −3.93725321871172263777531833941, −3.18023855880530203519615252116, −1.58743281801218509896114757290, 0, 1.58743281801218509896114757290, 3.18023855880530203519615252116, 3.93725321871172263777531833941, 4.33835034948233865658543563854, 4.97376859736062822975389284623, 5.83159640611586890531957946716, 6.83969790329135228911236082140, 7.25560493967332643536479769551, 7.85645423889849417776793402919, 8.062173761857703697875994376929, 8.751287964350617633590160637874, 9.136661245630303130527527106950, 9.950081302442658861535765863833, 10.19023449659800280025304267480

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