Properties

Label 2-1617-1.1-c3-0-135
Degree $2$
Conductor $1617$
Sign $-1$
Analytic cond. $95.4060$
Root an. cond. $9.76760$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.86·2-s − 3·3-s − 4.50·4-s + 9.74·5-s + 5.60·6-s + 23.3·8-s + 9·9-s − 18.2·10-s + 11·11-s + 13.5·12-s + 20.2·13-s − 29.2·15-s − 7.58·16-s − 15.3·17-s − 16.8·18-s − 26.0·19-s − 43.9·20-s − 20.5·22-s + 106.·23-s − 70.1·24-s − 30.0·25-s − 37.9·26-s − 27·27-s − 286.·29-s + 54.6·30-s − 120.·31-s − 172.·32-s + ⋯
L(s)  = 1  − 0.660·2-s − 0.577·3-s − 0.563·4-s + 0.871·5-s + 0.381·6-s + 1.03·8-s + 0.333·9-s − 0.575·10-s + 0.301·11-s + 0.325·12-s + 0.433·13-s − 0.503·15-s − 0.118·16-s − 0.218·17-s − 0.220·18-s − 0.315·19-s − 0.491·20-s − 0.199·22-s + 0.964·23-s − 0.596·24-s − 0.240·25-s − 0.286·26-s − 0.192·27-s − 1.83·29-s + 0.332·30-s − 0.697·31-s − 0.954·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(95.4060\)
Root analytic conductor: \(9.76760\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1617,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
7 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 1.86T + 8T^{2} \)
5 \( 1 - 9.74T + 125T^{2} \)
13 \( 1 - 20.2T + 2.19e3T^{2} \)
17 \( 1 + 15.3T + 4.91e3T^{2} \)
19 \( 1 + 26.0T + 6.85e3T^{2} \)
23 \( 1 - 106.T + 1.21e4T^{2} \)
29 \( 1 + 286.T + 2.43e4T^{2} \)
31 \( 1 + 120.T + 2.97e4T^{2} \)
37 \( 1 - 132.T + 5.06e4T^{2} \)
41 \( 1 - 130.T + 6.89e4T^{2} \)
43 \( 1 - 4.42T + 7.95e4T^{2} \)
47 \( 1 + 41.8T + 1.03e5T^{2} \)
53 \( 1 - 80.9T + 1.48e5T^{2} \)
59 \( 1 + 379.T + 2.05e5T^{2} \)
61 \( 1 - 25.0T + 2.26e5T^{2} \)
67 \( 1 + 137.T + 3.00e5T^{2} \)
71 \( 1 + 1.08e3T + 3.57e5T^{2} \)
73 \( 1 - 885.T + 3.89e5T^{2} \)
79 \( 1 + 65.3T + 4.93e5T^{2} \)
83 \( 1 - 823.T + 5.71e5T^{2} \)
89 \( 1 - 150.T + 7.04e5T^{2} \)
97 \( 1 - 53.8T + 9.12e5T^{2} \)
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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

−8.992833115321003703071440282107, −7.87552483927927498940319909485, −7.11745699163032272135867638581, −6.10845342876602353315963511183, −5.46361862804769096693685124813, −4.55556794534402263544538396711, −3.61997947984162767409256420432, −2.04285615376017000944507429992, −1.16663752618927025308289575133, 0, 1.16663752618927025308289575133, 2.04285615376017000944507429992, 3.61997947984162767409256420432, 4.55556794534402263544538396711, 5.46361862804769096693685124813, 6.10845342876602353315963511183, 7.11745699163032272135867638581, 7.87552483927927498940319909485, 8.992833115321003703071440282107

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