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Properties

Label 2-2312-136.43-c0-0-4
Degree $2$
Conductor $2312$
Sign $-0.651 + 0.758i$
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 − 1.00i·4-s + (−0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s − 1.00·16-s − 1.00·18-s + (1.41 − 1.41i)19-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + (−0.707 + 0.707i)36-s − 2.00i·38-s + (−1.41 − 1.41i)43-s + (−0.707 + 0.707i)49-s − 1.00·50-s + (1.41 + 1.41i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s − 1.00·16-s − 1.00·18-s + (1.41 − 1.41i)19-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + (−0.707 + 0.707i)36-s − 2.00i·38-s + (−1.41 − 1.41i)43-s + (−0.707 + 0.707i)49-s − 1.00·50-s + (1.41 + 1.41i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.651 + 0.758i$
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: \((2,\ 2312,\ (\ :0),\ -0.651 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.483805527\)
\(L(\frac12)\) \(\approx\) \(1.483805527\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−9.154841269806431373905495220659, −8.308535154181795719986657528786, −7.12451743548171411674074350248, −6.45342254332649190769180668322, −5.55420115214309402874914595729, −4.95445696055809479436942117576, −3.87443872781623881171309099983, −3.13336257552396451774035909162, −2.26494508970013943867371643918, −0.799591560594038470159497330029, 1.91442494301080118043109880529, 3.13538491372726399786320898136, 3.76389476188217843196402056899, 5.00542919460130982611186626472, 5.43560006535886436062470569220, 6.24645980435625160952427942308, 7.12598726578523667847234930753, 8.057010221187542199659873024117, 8.196092084624579612833726988498, 9.410314279702778333505706039580

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