Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} \cdot 13 \cdot 17^{2} $
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 − 3·8-s + 6·9-s + 13-s − 16-s + 2·17-s + 6·18-s + 25-s + 26-s + 5·32-s + 2·34-s − 6·36-s − 16·37-s − 2·49-s + 50-s − 52-s − 20·53-s + 16·61-s + 7·64-s − 2·68-s − 18·72-s + 16·73-s − 16·74-s + 27·81-s − 4·89-s − 2·98-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s + 2·9-s + 0.277·13-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 1/5·25-s + 0.196·26-s + 0.883·32-s + 0.342·34-s − 36-s − 2.63·37-s − 2/7·49-s + 0.141·50-s − 0.138·52-s − 2.74·53-s + 2.04·61-s + 7/8·64-s − 0.242·68-s − 2.12·72-s + 1.87·73-s − 1.85·74-s + 3·81-s − 0.423·89-s − 0.202·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1502800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1502800\)    =    \(2^{4} \cdot 5^{2} \cdot 13 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1502800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 1502800,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.850348184\)
\(L(\frac12)\)  \(\approx\)  \(2.850348184\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.956344814679173560130645750704, −7.42098416202486026176332520911, −6.88151949200386059528180448479, −6.64564283838235847728460954301, −6.24354007413923031454928173288, −5.53696407863215736767361553791, −5.13186745572228832228202757437, −4.82683455452161309392174542646, −4.35613449498539556535200530458, −3.77548766902101660819221290155, −3.54396306238639053983018236165, −3.00824810790453349077050332576, −2.01454927911837243427057514499, −1.56445010767842882439297787333, −0.68068940751151498273957179065, 0.68068940751151498273957179065, 1.56445010767842882439297787333, 2.01454927911837243427057514499, 3.00824810790453349077050332576, 3.54396306238639053983018236165, 3.77548766902101660819221290155, 4.35613449498539556535200530458, 4.82683455452161309392174542646, 5.13186745572228832228202757437, 5.53696407863215736767361553791, 6.24354007413923031454928173288, 6.64564283838235847728460954301, 6.88151949200386059528180448479, 7.42098416202486026176332520911, 7.956344814679173560130645750704

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