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Properties

Label 2-1617-77.76-c1-0-2
Degree $2$
Conductor $1617$
Sign $0.938 - 0.346i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
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  + 2.69i·2-s + i·3-s − 5.27·4-s + 3.60i·5-s − 2.69·6-s − 8.81i·8-s − 9-s − 9.72·10-s + (−2.32 − 2.36i)11-s − 5.27i·12-s − 4.29·13-s − 3.60·15-s + 13.2·16-s + 6.77·17-s − 2.69i·18-s − 2.03·19-s + ⋯
L(s)  = 1  + 1.90i·2-s + 0.577i·3-s − 2.63·4-s + 1.61i·5-s − 1.10·6-s − 3.11i·8-s − 0.333·9-s − 3.07·10-s + (−0.699 − 0.714i)11-s − 1.52i·12-s − 1.19·13-s − 0.931·15-s + 3.30·16-s + 1.64·17-s − 0.635i·18-s − 0.465·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.938 - 0.346i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 0.938 - 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1387767642\)
\(L(\frac12)\) \(\approx\) \(0.1387767642\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (2.32 + 2.36i)T \)
good2 \( 1 - 2.69iT - 2T^{2} \)
5 \( 1 - 3.60iT - 5T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
17 \( 1 - 6.77T + 17T^{2} \)
19 \( 1 + 2.03T + 19T^{2} \)
23 \( 1 - 1.20T + 23T^{2} \)
29 \( 1 - 0.635iT - 29T^{2} \)
31 \( 1 + 5.92iT - 31T^{2} \)
37 \( 1 + 9.93T + 37T^{2} \)
41 \( 1 - 1.61T + 41T^{2} \)
43 \( 1 - 6.25iT - 43T^{2} \)
47 \( 1 + 0.262iT - 47T^{2} \)
53 \( 1 + 7.12T + 53T^{2} \)
59 \( 1 - 3.79iT - 59T^{2} \)
61 \( 1 - 6.15T + 61T^{2} \)
67 \( 1 - 6.29T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 5.55T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + 1.67T + 83T^{2} \)
89 \( 1 + 8.75iT - 89T^{2} \)
97 \( 1 - 2.61iT - 97T^{2} \)
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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

−10.08604486759613023034060582393, −9.351097781322865617160701015864, −8.302333147885369681983699753805, −7.61746180809688419341957771842, −7.14032942447714797109801490532, −6.19227919163823536540055538969, −5.61615603691358422967772682774, −4.80243192535648594039167137056, −3.65709892920194214218011499311, −2.86867223193640428257862896631, 0.05861976037687260214465020840, 1.22663203104682702961709250027, 2.02166817393628100568139164523, 3.07653800467262940979727287670, 4.21066283053375834450759948439, 5.16135693127384831760254895109, 5.33236392549502623021932699478, 7.28106017035846464741378951228, 8.193868479260689268871862845200, 8.715110395504021613044240972689

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