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Properties

Label 2-1617-77.76-c1-0-60
Degree $2$
Conductor $1617$
Sign $-0.969 + 0.245i$
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  − 1.72i·2-s + i·3-s − 0.979·4-s + 1.65i·5-s + 1.72·6-s − 1.76i·8-s − 9-s + 2.85·10-s + (−2.71 − 1.89i)11-s − 0.979i·12-s + 2.97·13-s − 1.65·15-s − 4.99·16-s − 1.49·17-s + 1.72i·18-s − 5.51·19-s + ⋯
L(s)  = 1  − 1.22i·2-s + 0.577i·3-s − 0.489·4-s + 0.740i·5-s + 0.704·6-s − 0.622i·8-s − 0.333·9-s + 0.903·10-s + (−0.820 − 0.572i)11-s − 0.282i·12-s + 0.825·13-s − 0.427·15-s − 1.24·16-s − 0.362·17-s + 0.406i·18-s − 1.26·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.969 + 0.245i$
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: \((2,\ 1617,\ (\ :1/2),\ -0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9396870384\)
\(L(\frac12)\) \(\approx\) \(0.9396870384\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.71 + 1.89i)T \)
good2 \( 1 + 1.72iT - 2T^{2} \)
5 \( 1 - 1.65iT - 5T^{2} \)
13 \( 1 - 2.97T + 13T^{2} \)
17 \( 1 + 1.49T + 17T^{2} \)
19 \( 1 + 5.51T + 19T^{2} \)
23 \( 1 + 8.62T + 23T^{2} \)
29 \( 1 + 8.02iT - 29T^{2} \)
31 \( 1 + 8.86iT - 31T^{2} \)
37 \( 1 - 8.22T + 37T^{2} \)
41 \( 1 - 8.12T + 41T^{2} \)
43 \( 1 + 5.03iT - 43T^{2} \)
47 \( 1 + 7.36iT - 47T^{2} \)
53 \( 1 + 1.26T + 53T^{2} \)
59 \( 1 - 4.96iT - 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 - 9.85T + 73T^{2} \)
79 \( 1 + 3.93iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 5.16iT - 89T^{2} \)
97 \( 1 - 3.35iT - 97T^{2} \)
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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

−9.343194844746102259651960755782, −8.402957328362867450379635200333, −7.62383227784281909422740568788, −6.30123695416974217729109916045, −5.92109986351719385938337976464, −4.28639151955524488865641745449, −3.89976548535606301659271007774, −2.73322657566312930603938413870, −2.17667579560171458622184026137, −0.33952501662030287806166174668, 1.54183565707230993990598066961, 2.67253646774007919717206598613, 4.29311856551989919565196234287, 5.01993971664078034845972075625, 5.96537000854666038403508169571, 6.48949893962959288877842677862, 7.38211853921616663914994211635, 8.136780744591722775367308445690, 8.552998264301419195842196130771, 9.371137034630296624021074647507

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