Properties

Degree 4
Conductor $ 2^{5} \cdot 97 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 4·9-s − 2·10-s + 16-s + 17-s − 4·18-s − 2·20-s − 7·25-s + 11·29-s + 32-s + 34-s − 4·36-s − 8·37-s − 2·40-s + 7·41-s + 8·45-s + 8·49-s − 7·50-s + 3·53-s + 11·58-s + 5·61-s + 64-s + 68-s − 4·72-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 4/3·9-s − 0.632·10-s + 1/4·16-s + 0.242·17-s − 0.942·18-s − 0.447·20-s − 7/5·25-s + 2.04·29-s + 0.176·32-s + 0.171·34-s − 2/3·36-s − 1.31·37-s − 0.316·40-s + 1.09·41-s + 1.19·45-s + 8/7·49-s − 0.989·50-s + 0.412·53-s + 1.44·58-s + 0.640·61-s + 1/8·64-s + 0.121·68-s − 0.471·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3104\)    =    \(2^{5} \cdot 97\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3104} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 3104,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9020190524$
$L(\frac12)$  $\approx$  $0.9020190524$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;97\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;97\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 - T \)
97$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 14 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.59553291146915511080661092877, −12.14914357965763811402258945951, −11.67666452745737778737148941465, −11.28615733627910955502391790681, −10.51284626564333131231254367811, −9.918635681762329691135585363506, −8.864568291064431286214969772691, −8.368363599824135263311855725638, −7.72105610635250465375639522353, −6.98295168141343444896521569686, −6.05386674990596023650926542265, −5.48493353587528670096291773113, −4.46988023346650659116902833531, −3.62568588139740439126212134267, −2.64765870112626908625429132935, 2.64765870112626908625429132935, 3.62568588139740439126212134267, 4.46988023346650659116902833531, 5.48493353587528670096291773113, 6.05386674990596023650926542265, 6.98295168141343444896521569686, 7.72105610635250465375639522353, 8.368363599824135263311855725638, 8.864568291064431286214969772691, 9.918635681762329691135585363506, 10.51284626564333131231254367811, 11.28615733627910955502391790681, 11.67666452745737778737148941465, 12.14914357965763811402258945951, 12.59553291146915511080661092877

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