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Properties

Label 2-2312-136.43-c0-0-3
Degree $2$
Conductor $2312$
Sign $0.641 + 0.766i$
Analytic cond. $1.15383$
Root an. cond. $1.07416$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (1.30 + 0.541i)3-s − 1.00i·4-s + (1.30 − 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (1.30 − 0.541i)11-s + (0.541 − 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (0.541 − 1.30i)22-s + (−0.541 − 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (1.30 + 0.541i)3-s − 1.00i·4-s + (1.30 − 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (1.30 − 0.541i)11-s + (0.541 − 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (0.541 − 1.30i)22-s + (−0.541 − 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.641 + 0.766i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.641 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.609659361\)
\(L(\frac12)\) \(\approx\) \(2.609659361\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.191552804832377898369248086647, −8.530642612055003120456434045981, −7.81165444107721647012663646206, −6.39594923942140469639928482294, −6.04835239815721297707047084873, −4.64012955515008919082373011073, −3.96838246723757214886267187142, −3.47749074301557145404796958707, −2.47319277334876745834661721170, −1.54592500021684048866242602610, 1.86492933360022008569615523867, 2.66196805033904237836773195518, 3.72466805569374275558531690887, 4.27295540831843669906119754561, 5.35776477614645538422072039270, 6.52892305889385808465772689704, 6.92578894769983584778785854646, 7.66182035680228842063539719853, 8.468834279353769589673102865771, 9.014097593574798431248703782578

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