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Properties

Label 2-2312-136.123-c0-0-3
Degree $2$
Conductor $2312$
Sign $-0.933 + 0.359i$
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  − i·2-s + (−0.541 + 0.541i)3-s − 4-s + (0.541 + 0.541i)6-s + i·8-s + 0.414i·9-s + (−1.30 − 1.30i)11-s + (0.541 − 0.541i)12-s + 16-s + 0.414·18-s + (−1.30 + 1.30i)22-s + (−0.541 − 0.541i)24-s − i·25-s + (−0.765 − 0.765i)27-s − i·32-s + 1.41·33-s + ⋯
L(s)  = 1  − i·2-s + (−0.541 + 0.541i)3-s − 4-s + (0.541 + 0.541i)6-s + i·8-s + 0.414i·9-s + (−1.30 − 1.30i)11-s + (0.541 − 0.541i)12-s + 16-s + 0.414·18-s + (−1.30 + 1.30i)22-s + (−0.541 − 0.541i)24-s − i·25-s + (−0.765 − 0.765i)27-s − i·32-s + 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.933 + 0.359i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ -0.933 + 0.359i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4236086069\)
\(L(\frac12)\) \(\approx\) \(0.4236086069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.909802641969591209469861060486, −8.291585116365379019961611362546, −7.60284924491217834412456050094, −6.16601849857377489757758693999, −5.38985583717774992468163427373, −4.89779962029686630133784089501, −3.88812081554661159980427203610, −2.99174891031813312693217835031, −2.05945636248907190596913120402, −0.31573344612786096436728601680, 1.44934770163841037696702416150, 2.97468205553403247791573493656, 4.18829200790195373977773148518, 5.07223166618728999105609005340, 5.63196848368868982128936357205, 6.59317886440716671591402050451, 7.12750089655354293373044858659, 7.76872546513392089816590923768, 8.516002113609546780279108163833, 9.579074827234737442791920022633

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