Properties

Degree 4
Conductor $ 2^{5} \cdot 53 $
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 − 6·5-s + 8-s + 9-s − 6·10-s + 13-s + 16-s + 18-s − 6·20-s + 17·25-s + 26-s − 3·29-s + 32-s + 36-s + 4·37-s − 6·40-s + 3·41-s − 6·45-s + 5·49-s + 17·50-s + 52-s − 53-s − 3·58-s − 11·61-s + 64-s − 6·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 2.68·5-s + 0.353·8-s + 1/3·9-s − 1.89·10-s + 0.277·13-s + 1/4·16-s + 0.235·18-s − 1.34·20-s + 17/5·25-s + 0.196·26-s − 0.557·29-s + 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.948·40-s + 0.468·41-s − 0.894·45-s + 5/7·49-s + 2.40·50-s + 0.138·52-s − 0.137·53-s − 0.393·58-s − 1.40·61-s + 1/8·64-s − 0.744·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1696\)    =    \(2^{5} \cdot 53\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1696,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.6650381480$
$L(\frac12)$  $\approx$  $0.6650381480$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;53\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;53\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
53$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$V_4$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$V_4$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$V_4$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$V_4$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$V_4$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
47$V_4$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
59$V_4$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
71$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$V_4$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−13.42724682093127389972172069157, −12.78145559758971114227274539158, −12.13596521781658862122727281889, −11.87390556995695397480803211808, −11.08479540647462308328149105150, −10.91351350997036117749863180427, −9.778045086497591874366562288401, −8.730377005987351492684344960944, −8.043431431153194891349173259256, −7.50545797123803551468017616240, −6.99213001189050982833153591153, −5.83022377552465209703060643418, −4.53328554630915693594150669766, −4.07419463302334338117869636017, −3.22192610779849948278652519682, 3.22192610779849948278652519682, 4.07419463302334338117869636017, 4.53328554630915693594150669766, 5.83022377552465209703060643418, 6.99213001189050982833153591153, 7.50545797123803551468017616240, 8.043431431153194891349173259256, 8.730377005987351492684344960944, 9.778045086497591874366562288401, 10.91351350997036117749863180427, 11.08479540647462308328149105150, 11.87390556995695397480803211808, 12.13596521781658862122727281889, 12.78145559758971114227274539158, 13.42724682093127389972172069157

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