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Properties

Label 2-2312-136.123-c0-0-2
Degree $2$
Conductor $2312$
Sign $0.432 + 0.901i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.432 + 0.901i$
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: \((2,\ 2312,\ (\ :0),\ 0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.376595943\)
\(L(\frac12)\) \(\approx\) \(1.376595943\)
\(L(1)\) 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

−9.131210063449196782888057400312, −8.441853469153607616432371556300, −7.64403332309484435477072835107, −6.91427163985877113242488297098, −5.87685749251434964564817038165, −4.65817679056871783721308987534, −4.20466420404829451237249690744, −3.02234452380382829254359401375, −2.13356710857894642333339238677, −1.37550194645472973277388915930, 1.09959386868838736350863992907, 3.07635294742206398525582105559, 3.77631790482260275895402657340, 4.42123760656916421123364773835, 5.62040753298722902376367288375, 6.18694868864684843277742813315, 6.97149808779255547877968584067, 7.83930588054371194550718330935, 8.807481563150045922156290456370, 8.991792620688945131920607885205

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