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Properties

Label 2-2312-136.67-c0-0-1
Degree $2$
Conductor $2312$
Sign $-0.443 - 0.896i$
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 + 1.84i·3-s + 4-s + 1.84i·6-s + 8-s − 2.41·9-s + 0.765i·11-s + 1.84i·12-s + 16-s − 2.41·18-s + 0.765i·22-s + 1.84i·24-s − 25-s − 2.61i·27-s + 32-s − 1.41·33-s + ⋯
L(s)  = 1  + 2-s + 1.84i·3-s + 4-s + 1.84i·6-s + 8-s − 2.41·9-s + 0.765i·11-s + 1.84i·12-s + 16-s − 2.41·18-s + 0.765i·22-s + 1.84i·24-s − 25-s − 2.61i·27-s + 32-s − 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.443 - 0.896i$
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: \((2,\ 2312,\ (\ :0),\ -0.443 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.154920831\)
\(L(\frac12)\) \(\approx\) \(2.154920831\)
\(L(1)\) 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 \( 1 - 1.84iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 0.765iT - T^{2} \)
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 \( 1 + 0.765iT - T^{2} \)
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 \( 1 + 0.765iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−9.799074766827729766719975922139, −8.817351389388166227078759153314, −7.916350101866167997253455298910, −6.94719658859072979574185010542, −5.87258940131523366281139428756, −5.33954197312412410385418610319, −4.47307327880330739163961748106, −4.01327787892216199240596307387, −3.16508919462040844262041376144, −2.16888566552807784737086942249, 1.11513564146595819812379210394, 2.16685111938974556394570715268, 2.93407343565805632459760655764, 3.95305571431710732757907521244, 5.30201099008210202882970595301, 5.91587376466374908766782748859, 6.54913519797442698566002083339, 7.22340460843445517885787346151, 7.959987185144683680824333149333, 8.470348655253670647127176923839

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