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Properties

Label 2-2312-136.67-c0-0-2
Degree $2$
Conductor $2312$
Sign $0.928 + 0.371i$
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  + 2-s − 0.765i·3-s + 4-s − 0.765i·6-s + 8-s + 0.414·9-s + 1.84i·11-s − 0.765i·12-s + 16-s + 0.414·18-s + 1.84i·22-s − 0.765i·24-s − 25-s − 1.08i·27-s + 32-s + 1.41·33-s + ⋯
L(s)  = 1  + 2-s − 0.765i·3-s + 4-s − 0.765i·6-s + 8-s + 0.414·9-s + 1.84i·11-s − 0.765i·12-s + 16-s + 0.414·18-s + 1.84i·22-s − 0.765i·24-s − 25-s − 1.08i·27-s + 32-s + 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.928 + 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.385004575\)
\(L(\frac12)\) \(\approx\) \(2.385004575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
17 \( 1 \)
good3 \( 1 + 0.765iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.84iT - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.84iT - T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.84iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - 0.765iT - T^{2} \)
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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

−9.229218167994540509153750067043, −7.85386848797870802125537994605, −7.48719809043220228787416153086, −6.77498810149992700099243065840, −6.14246849915926341256754206949, −5.02746312591633334483253417758, −4.43569063839320089468878381742, −3.49735471448734578166262831048, −2.16163862277811396803170736173, −1.68736117982966711538729898723, 1.50334879127111287801482327412, 2.98850992480236095463409839170, 3.54502636005890350720067784769, 4.40836613240779100121196930970, 5.17546102674073364881760987646, 5.99694088359470745665863546721, 6.59105462325326269987935454796, 7.70490558494128723184444323180, 8.346326027735914529673091643165, 9.372482877756690138035207819556

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