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Properties

Label 2-1617-77.76-c1-0-19
Degree $2$
Conductor $1617$
Sign $-0.902 + 0.431i$
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  + 1.47i·2-s + i·3-s − 0.170·4-s + 1.04i·5-s − 1.47·6-s + 2.69i·8-s − 9-s − 1.53·10-s + (2.52 + 2.14i)11-s − 0.170i·12-s + 0.300·13-s − 1.04·15-s − 4.31·16-s − 1.19·17-s − 1.47i·18-s − 3.70·19-s + ⋯
L(s)  = 1  + 1.04i·2-s + 0.577i·3-s − 0.0851·4-s + 0.466i·5-s − 0.601·6-s + 0.953i·8-s − 0.333·9-s − 0.486·10-s + (0.762 + 0.646i)11-s − 0.0491i·12-s + 0.0832·13-s − 0.269·15-s − 1.07·16-s − 0.289·17-s − 0.347i·18-s − 0.849·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.902 + 0.431i$
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: \((2,\ 1617,\ (\ :1/2),\ -0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.603193644\)
\(L(\frac12)\) \(\approx\) \(1.603193644\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 - 2.14i)T \)
good2 \( 1 - 1.47iT - 2T^{2} \)
5 \( 1 - 1.04iT - 5T^{2} \)
13 \( 1 - 0.300T + 13T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 + 3.70T + 19T^{2} \)
23 \( 1 + 0.990T + 23T^{2} \)
29 \( 1 - 6.66iT - 29T^{2} \)
31 \( 1 - 0.951iT - 31T^{2} \)
37 \( 1 + 5.94T + 37T^{2} \)
41 \( 1 + 0.780T + 41T^{2} \)
43 \( 1 + 1.86iT - 43T^{2} \)
47 \( 1 - 6.34iT - 47T^{2} \)
53 \( 1 - 9.54T + 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + 5.58T + 61T^{2} \)
67 \( 1 - 2.89T + 67T^{2} \)
71 \( 1 - 7.01T + 71T^{2} \)
73 \( 1 + 4.39T + 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 + 17.6T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 - 0.642iT - 97T^{2} \)
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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

−9.724784305210015782618759656187, −8.821733784075057927001428406885, −8.327067906146900860708147850966, −7.09488447330012927145753434961, −6.83185707439505528953225045023, −5.92596824429520008102119081478, −5.02926616790189731122916733330, −4.20756523836826983574972206368, −3.06337129183376626427901811409, −1.88299362025515411221679837398, 0.59822230971859231202250366246, 1.65284100135157276444574448573, 2.59413232302131531322486170861, 3.65806058312606928775002239768, 4.46484420435394650562800862521, 5.78141538728887652585344233195, 6.53660062903533011408710999755, 7.26207741915133407215694267613, 8.442780814301355717074309499702, 8.907338276405356276087384039919

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