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Properties

Label 2-2312-17.16-c1-0-42
Degree $2$
Conductor $2312$
Sign $0.743 - 0.669i$
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  + 2.18i·3-s + 1.65i·5-s − 3.87i·7-s − 1.77·9-s + 1.06i·11-s + 6.10·13-s − 3.61·15-s − 1.41·19-s + 8.47·21-s − 6.92i·23-s + 2.26·25-s + 2.68i·27-s − 7.90i·29-s − 9.00i·31-s − 2.32·33-s + ⋯
L(s)  = 1  + 1.26i·3-s + 0.739i·5-s − 1.46i·7-s − 0.591·9-s + 0.320i·11-s + 1.69·13-s − 0.932·15-s − 0.323·19-s + 1.84·21-s − 1.44i·23-s + 0.453·25-s + 0.515i·27-s − 1.46i·29-s − 1.61i·31-s − 0.404·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.743 - 0.669i$
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: \((2,\ 2312,\ (\ :1/2),\ 0.743 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987902251\)
\(L(\frac12)\) \(\approx\) \(1.987902251\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.18iT - 3T^{2} \)
5 \( 1 - 1.65iT - 5T^{2} \)
7 \( 1 + 3.87iT - 7T^{2} \)
11 \( 1 - 1.06iT - 11T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
19 \( 1 + 1.41T + 19T^{2} \)
23 \( 1 + 6.92iT - 23T^{2} \)
29 \( 1 + 7.90iT - 29T^{2} \)
31 \( 1 + 9.00iT - 31T^{2} \)
37 \( 1 - 3.47iT - 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 8.86T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 - 1.06T + 53T^{2} \)
59 \( 1 - 9.46T + 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 - 2.04T + 67T^{2} \)
71 \( 1 - 0.448iT - 71T^{2} \)
73 \( 1 + 1.83iT - 73T^{2} \)
79 \( 1 + 2.75iT - 79T^{2} \)
83 \( 1 + 0.389T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 13.7iT - 97T^{2} \)
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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

−9.310544869459643471815353879763, −8.322261815807397291404039816681, −7.63009138900435730542064962839, −6.59057934632167207224543769738, −6.15347231169531288358307883493, −4.79355088957534373612187456353, −4.06428456325102454269720290872, −3.73786566694440881131573840508, −2.55219731630794553181542048121, −0.879729574476115711818081898367, 1.07371600414511898298444720613, 1.76905342707560849878614089291, 2.89597876458962875206005376078, 3.94846222942019015260171543561, 5.45098987797341436612405482675, 5.62135185745182336445368188982, 6.61677444548919214473570304531, 7.31765169029367688582524051207, 8.382286968233322134853061825758, 8.753599293184982358640239303142

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