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Properties

Label 2-2312-17.16-c1-0-3
Degree $2$
Conductor $2312$
Sign $0.168 + 0.985i$
Analytic cond. $18.4614$
Root an. cond. $4.29667$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.22i·3-s + 3.87i·5-s − 0.467i·7-s − 7.41·9-s − 1.30i·11-s − 2.94·13-s − 12.5·15-s + 3.18·19-s + 1.50·21-s + 3.17i·23-s − 10.0·25-s − 14.2i·27-s − 8.80i·29-s − 0.857i·31-s + 4.21·33-s + ⋯
L(s)  = 1  + 1.86i·3-s + 1.73i·5-s − 0.176i·7-s − 2.47·9-s − 0.393i·11-s − 0.816·13-s − 3.23·15-s + 0.730·19-s + 0.329·21-s + 0.661i·23-s − 2.00·25-s − 2.73i·27-s − 1.63i·29-s − 0.153i·31-s + 0.733·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(18.4614\)
Root analytic conductor: \(4.29667\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :1/2),\ 0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6446763034\)
\(L(\frac12)\) \(\approx\) \(0.6446763034\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 - 3.22iT - 3T^{2} \)
5 \( 1 - 3.87iT - 5T^{2} \)
7 \( 1 + 0.467iT - 7T^{2} \)
11 \( 1 + 1.30iT - 11T^{2} \)
13 \( 1 + 2.94T + 13T^{2} \)
19 \( 1 - 3.18T + 19T^{2} \)
23 \( 1 - 3.17iT - 23T^{2} \)
29 \( 1 + 8.80iT - 29T^{2} \)
31 \( 1 + 0.857iT - 31T^{2} \)
37 \( 1 + 3.49iT - 37T^{2} \)
41 \( 1 - 8.12iT - 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 + 2.41T + 47T^{2} \)
53 \( 1 + 1.30T + 53T^{2} \)
59 \( 1 + 9.96T + 59T^{2} \)
61 \( 1 - 3.35iT - 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + 5.10iT - 73T^{2} \)
79 \( 1 - 4.06iT - 79T^{2} \)
83 \( 1 - 8.51T + 83T^{2} \)
89 \( 1 + 4.31T + 89T^{2} \)
97 \( 1 - 6.93iT - 97T^{2} \)
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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

−9.827811484004542507849591000140, −9.192749682567594107684987668889, −8.042761629549631111510307585106, −7.36568330655436683060040249849, −6.29455080500618563621174088930, −5.65952276291610115235114360463, −4.69621395600370110307686297038, −3.83017755244629301557110392961, −3.16021848145932035554002982215, −2.50057396235909439323765279009, 0.21921036058206667194632426012, 1.30308764294944402021288938279, 1.97373790637667700778012902648, 3.15038598475031584103172944367, 4.71605878493975489416252885395, 5.26671364629412090300613250858, 6.08999499575502172502023739302, 7.06419542437279209131697171900, 7.57366585983047199252230400391, 8.408471983458016606403867971761

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