Properties

Label 2-1617-1.1-c3-0-190
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  + 4.50·2-s − 3·3-s + 12.2·4-s + 3.51·5-s − 13.5·6-s + 19.1·8-s + 9·9-s + 15.8·10-s + 11·11-s − 36.7·12-s − 4.32·13-s − 10.5·15-s − 11.7·16-s − 87.2·17-s + 40.5·18-s − 70.0·19-s + 43.1·20-s + 49.5·22-s + 104.·23-s − 57.5·24-s − 112.·25-s − 19.4·26-s − 27·27-s + 8.09·29-s − 47.4·30-s − 79.0·31-s − 206.·32-s + ⋯
L(s)  = 1  + 1.59·2-s − 0.577·3-s + 1.53·4-s + 0.314·5-s − 0.918·6-s + 0.847·8-s + 0.333·9-s + 0.500·10-s + 0.301·11-s − 0.884·12-s − 0.0922·13-s − 0.181·15-s − 0.183·16-s − 1.24·17-s + 0.530·18-s − 0.846·19-s + 0.482·20-s + 0.479·22-s + 0.950·23-s − 0.489·24-s − 0.901·25-s − 0.146·26-s − 0.192·27-s + 0.0518·29-s − 0.289·30-s − 0.458·31-s − 1.13·32-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 \( 1 - 4.50T + 8T^{2} \)
5 \( 1 - 3.51T + 125T^{2} \)
13 \( 1 + 4.32T + 2.19e3T^{2} \)
17 \( 1 + 87.2T + 4.91e3T^{2} \)
19 \( 1 + 70.0T + 6.85e3T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 - 8.09T + 2.43e4T^{2} \)
31 \( 1 + 79.0T + 2.97e4T^{2} \)
37 \( 1 - 24.4T + 5.06e4T^{2} \)
41 \( 1 + 436.T + 6.89e4T^{2} \)
43 \( 1 - 339.T + 7.95e4T^{2} \)
47 \( 1 - 164.T + 1.03e5T^{2} \)
53 \( 1 + 289.T + 1.48e5T^{2} \)
59 \( 1 - 49.4T + 2.05e5T^{2} \)
61 \( 1 + 411.T + 2.26e5T^{2} \)
67 \( 1 + 189.T + 3.00e5T^{2} \)
71 \( 1 + 138.T + 3.57e5T^{2} \)
73 \( 1 + 704.T + 3.89e5T^{2} \)
79 \( 1 - 988.T + 4.93e5T^{2} \)
83 \( 1 + 474.T + 5.71e5T^{2} \)
89 \( 1 - 443.T + 7.04e5T^{2} \)
97 \( 1 - 147.T + 9.12e5T^{2} \)
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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

−8.702409057925608289155500026673, −7.39216899384668667221495812906, −6.60193005669396356356714940403, −6.09430028477926707339589354186, −5.23485939900082558026231010154, −4.51704839943199551980714661149, −3.81081412282583360252378373410, −2.66151057390257169912536637784, −1.71153261040758338829610635148, 0, 1.71153261040758338829610635148, 2.66151057390257169912536637784, 3.81081412282583360252378373410, 4.51704839943199551980714661149, 5.23485939900082558026231010154, 6.09430028477926707339589354186, 6.60193005669396356356714940403, 7.39216899384668667221495812906, 8.702409057925608289155500026673

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