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Properties

Label 2-2312-136.19-c0-0-5
Degree $2$
Conductor $2312$
Sign $0.197 + 0.980i$
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 + (1.70 − 0.707i)3-s + 1.00i·4-s + (−1.70 − 0.707i)6-s + (0.707 − 0.707i)8-s + (1.70 − 1.70i)9-s + (0.707 + 0.292i)11-s + (0.707 + 1.70i)12-s − 1.00·16-s − 2.41·18-s + (−0.292 − 0.707i)22-s + (0.707 − 1.70i)24-s + (−0.707 + 0.707i)25-s + (1 − 2.41i)27-s + (0.707 + 0.707i)32-s + 1.41·33-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.70 − 0.707i)3-s + 1.00i·4-s + (−1.70 − 0.707i)6-s + (0.707 − 0.707i)8-s + (1.70 − 1.70i)9-s + (0.707 + 0.292i)11-s + (0.707 + 1.70i)12-s − 1.00·16-s − 2.41·18-s + (−0.292 − 0.707i)22-s + (0.707 − 1.70i)24-s + (−0.707 + 0.707i)25-s + (1 − 2.41i)27-s + (0.707 + 0.707i)32-s + 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.197 + 0.980i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.571821570\)
\(L(\frac12)\) \(\approx\) \(1.571821570\)
\(L(1)\) 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 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
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 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1 + i)T - iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−8.890072064139182856523573310264, −8.447135252715430213826434114060, −7.68534878569063562399281703497, −7.12835295941193555846896152249, −6.34637614902398797808182370126, −4.59098983523544564650247213987, −3.65224399183689659399527434535, −3.08410673378737922795376858486, −2.04232486294526781223947110950, −1.37761334328459750343212868521, 1.60543567120606853694421140740, 2.58272514617523842700937268225, 3.70656030308122473694639630299, 4.42680192249420259974834607908, 5.42563285977996062016900677346, 6.51723127441433228835649692264, 7.33087842898893887459740485987, 8.052315243573968231543283636121, 8.644220759694103909520191606888, 9.131558060590437169432217324578

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