Properties

Degree $4$
Conductor $2312$
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 − 2·9-s − 4·13-s − 16-s + 2·17-s − 2·18-s + 8·19-s − 6·25-s − 4·26-s + 5·32-s + 2·34-s + 2·36-s + 8·38-s + 8·47-s − 2·49-s − 6·50-s + 4·52-s − 4·53-s + 7·64-s − 8·67-s − 2·68-s + 6·72-s − 8·76-s − 5·81-s + 16·83-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2/3·9-s − 1.10·13-s − 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.83·19-s − 6/5·25-s − 0.784·26-s + 0.883·32-s + 0.342·34-s + 1/3·36-s + 1.29·38-s + 1.16·47-s − 2/7·49-s − 0.848·50-s + 0.554·52-s − 0.549·53-s + 7/8·64-s − 0.977·67-s − 0.242·68-s + 0.707·72-s − 0.917·76-s − 5/9·81-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2312} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7694790950\)
\(L(\frac12)\) \(\approx\) \(0.7694790950\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−13.29480901314820395116553055383, −12.47860606469799974443984036732, −11.93259226357823232140431724447, −11.72832068978073525243612985214, −10.75100662817001349923979099653, −9.789115370436604117705818298937, −9.552128918559945705841722441351, −8.781803539780993843603843763167, −7.86908856597964737620953720124, −7.36547148424824287699212207874, −6.17653615753153086734821876420, −5.47448849864809007342052544464, −4.90526824858536973933461518159, −3.77471359399535732815739676770, −2.82633709454958812603882709287, 2.82633709454958812603882709287, 3.77471359399535732815739676770, 4.90526824858536973933461518159, 5.47448849864809007342052544464, 6.17653615753153086734821876420, 7.36547148424824287699212207874, 7.86908856597964737620953720124, 8.781803539780993843603843763167, 9.552128918559945705841722441351, 9.789115370436604117705818298937, 10.75100662817001349923979099653, 11.72832068978073525243612985214, 11.93259226357823232140431724447, 12.47860606469799974443984036732, 13.29480901314820395116553055383

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