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Properties

Label 2-2312-17.16-c1-0-19
Degree $2$
Conductor $2312$
Sign $-0.928 - 0.371i$
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  + 1.84i·3-s + 0.765i·5-s + 4.46i·7-s − 0.414·9-s − 1.84i·11-s − 1.41·15-s + 6.24·19-s − 8.24·21-s + 4.90i·23-s + 4.41·25-s + 4.77i·27-s + 2.29i·29-s + 5.54i·31-s + 3.41·33-s − 3.41·35-s + ⋯
L(s)  = 1  + 1.06i·3-s + 0.342i·5-s + 1.68i·7-s − 0.138·9-s − 0.557i·11-s − 0.365·15-s + 1.43·19-s − 1.79·21-s + 1.02i·23-s + 0.882·25-s + 0.919i·27-s + 0.426i·29-s + 0.995i·31-s + 0.594·33-s − 0.577·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.928 - 0.371i$
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: \((2,\ 2312,\ (\ :1/2),\ -0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763416771\)
\(L(\frac12)\) \(\approx\) \(1.763416771\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.84iT - 3T^{2} \)
5 \( 1 - 0.765iT - 5T^{2} \)
7 \( 1 - 4.46iT - 7T^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 - 4.90iT - 23T^{2} \)
29 \( 1 - 2.29iT - 29T^{2} \)
31 \( 1 - 5.54iT - 31T^{2} \)
37 \( 1 + 1.84iT - 37T^{2} \)
41 \( 1 - 9.68iT - 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 + 2.48iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + 9.65T + 89T^{2} \)
97 \( 1 + 2.03iT - 97T^{2} \)
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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

−9.496954805749674171441270026962, −8.752992933994255810224283364205, −8.005733413383609408817621147592, −6.95217517408932309786864442775, −6.02624115327736124281743969712, −5.25091959417713687167516158780, −4.79236811647488140999428818976, −3.27400789438699735421586502755, −3.11697860514242253552606191941, −1.61355261222329820353182315669, 0.66195738697873842237529173407, 1.36291722421405065695845155632, 2.59915947265925990762667128948, 3.88593046390083072635768837373, 4.53752338596014859818246033808, 5.55628350875425261166124568881, 6.81456748594569811862952194597, 6.98533430132177888699340455263, 7.76785510976666045193960472224, 8.361699638942018944419862577609

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