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Properties

Label 2-2312-136.5-c0-0-1
Degree $2$
Conductor $2312$
Sign $0.518 - 0.855i$
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  + (0.382 + 0.923i)2-s + (1.17 − 0.785i)3-s + (−0.707 + 0.707i)4-s + (1.38 + 0.275i)5-s + (1.17 + 0.785i)6-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (0.275 + 1.38i)10-s + (−0.785 + 1.17i)11-s + (−0.275 + 1.38i)12-s + (1.84 − 0.765i)15-s − i·16-s + 0.999·18-s + (−1.17 + 0.785i)20-s + (−1.38 − 0.275i)22-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (1.17 − 0.785i)3-s + (−0.707 + 0.707i)4-s + (1.38 + 0.275i)5-s + (1.17 + 0.785i)6-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (0.275 + 1.38i)10-s + (−0.785 + 1.17i)11-s + (−0.275 + 1.38i)12-s + (1.84 − 0.765i)15-s − i·16-s + 0.999·18-s + (−1.17 + 0.785i)20-s + (−1.38 − 0.275i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (2181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.518 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.306372143\)
\(L(\frac12)\) \(\approx\) \(2.306372143\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 \)
good3 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
5 \( 1 + (-1.38 - 0.275i)T + (0.923 + 0.382i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.785 - 1.17i)T + (-0.382 - 0.923i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (1.17 - 0.785i)T + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.275 + 1.38i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−9.135826696886198646918628365804, −8.328321751155888364407285402682, −7.74142881731795475517861371260, −6.96098753728167604331742835022, −6.44857568105983151013027757626, −5.47160172116228783670485803694, −4.74511586954859605815273635967, −3.46743140483254534398452339284, −2.50710377872369727617862136820, −1.88093291829976715953917198729, 1.45916208903649324265388859610, 2.55521073486427893356738208684, 3.07736970617101001118486075063, 3.97400967059642687903358709294, 5.05319523378446415595122120127, 5.55359206591662604701485106512, 6.45126355980711887035498171645, 7.941973546327452496632753182615, 8.792009756373349964182727723365, 9.103723854096230461502628607769

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