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Properties

Label 2-2312-17.16-c1-0-49
Degree $2$
Conductor $2312$
Sign $-0.242 + 0.970i$
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  − 1.23i·3-s − 2i·5-s − 1.23i·7-s + 1.47·9-s − 1.23i·11-s + 4.47·13-s − 2.47·15-s + 6.47·19-s − 1.52·21-s − 1.23i·23-s + 25-s − 5.52i·27-s − 2i·29-s − 1.23i·31-s − 1.52·33-s + ⋯
L(s)  = 1  − 0.713i·3-s − 0.894i·5-s − 0.467i·7-s + 0.490·9-s − 0.372i·11-s + 1.24·13-s − 0.638·15-s + 1.48·19-s − 0.333·21-s − 0.257i·23-s + 0.200·25-s − 1.06i·27-s − 0.371i·29-s − 0.222i·31-s − 0.265·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 2312,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.156973500\)
\(L(\frac12)\) \(\approx\) \(2.156973500\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23iT - 3T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 1.23iT - 7T^{2} \)
11 \( 1 + 1.23iT - 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 1.23iT - 31T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 - 2iT - 41T^{2} \)
43 \( 1 - 1.52T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 - 6.94iT - 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 9.23iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + 7.52T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.733121376569298742040037696240, −7.929199226213651357722233009250, −7.36739930185601829855464292501, −6.44408920293976260199534288345, −5.75085289679847016066129862572, −4.74186876208367793484116591214, −3.97330543424891256394238101206, −2.90676796958382338734741811756, −1.38829159898304098553677403649, −0.923380169337622076423017427992, 1.36033899085931100334248189534, 2.69202596656722460830980195719, 3.54540783666180772382983142872, 4.24896984435959718763674588012, 5.37072696459483372389909542146, 5.96244761194600279784179001320, 7.10606131460798950382883474901, 7.43052138478424906179446927523, 8.687818083052977727114978119512, 9.266189411236188945847475095568

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