Properties

Label 2-1617-1.1-c3-0-107
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  + 2.12·2-s − 3·3-s − 3.49·4-s − 16.9·5-s − 6.36·6-s − 24.3·8-s + 9·9-s − 35.9·10-s + 11·11-s + 10.4·12-s + 28.2·13-s + 50.8·15-s − 23.8·16-s + 27.5·17-s + 19.1·18-s − 110.·19-s + 59.2·20-s + 23.3·22-s + 93.4·23-s + 73.1·24-s + 162.·25-s + 59.9·26-s − 27·27-s + 36.5·29-s + 107.·30-s + 210.·31-s + 144.·32-s + ⋯
L(s)  = 1  + 0.750·2-s − 0.577·3-s − 0.436·4-s − 1.51·5-s − 0.433·6-s − 1.07·8-s + 0.333·9-s − 1.13·10-s + 0.301·11-s + 0.252·12-s + 0.602·13-s + 0.875·15-s − 0.372·16-s + 0.392·17-s + 0.250·18-s − 1.33·19-s + 0.662·20-s + 0.226·22-s + 0.847·23-s + 0.622·24-s + 1.29·25-s + 0.452·26-s − 0.192·27-s + 0.234·29-s + 0.656·30-s + 1.21·31-s + 0.798·32-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 \( 1 - 2.12T + 8T^{2} \)
5 \( 1 + 16.9T + 125T^{2} \)
13 \( 1 - 28.2T + 2.19e3T^{2} \)
17 \( 1 - 27.5T + 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
23 \( 1 - 93.4T + 1.21e4T^{2} \)
29 \( 1 - 36.5T + 2.43e4T^{2} \)
31 \( 1 - 210.T + 2.97e4T^{2} \)
37 \( 1 - 421.T + 5.06e4T^{2} \)
41 \( 1 - 14.6T + 6.89e4T^{2} \)
43 \( 1 + 481.T + 7.95e4T^{2} \)
47 \( 1 + 287.T + 1.03e5T^{2} \)
53 \( 1 - 630.T + 1.48e5T^{2} \)
59 \( 1 + 587.T + 2.05e5T^{2} \)
61 \( 1 + 80.3T + 2.26e5T^{2} \)
67 \( 1 - 291.T + 3.00e5T^{2} \)
71 \( 1 + 114.T + 3.57e5T^{2} \)
73 \( 1 + 54.3T + 3.89e5T^{2} \)
79 \( 1 - 739.T + 4.93e5T^{2} \)
83 \( 1 + 110.T + 5.71e5T^{2} \)
89 \( 1 - 268.T + 7.04e5T^{2} \)
97 \( 1 + 1.52e3T + 9.12e5T^{2} \)
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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

−8.434730402021773477289920520526, −7.976041230983647907367889184985, −6.76924117660136728088581483024, −6.19744060196011251760500061999, −5.08288345322867528329038867526, −4.37283475175060562443840853129, −3.83529311116720005700264068500, −2.89744941776297393731018950688, −0.986399814140447695002969299084, 0, 0.986399814140447695002969299084, 2.89744941776297393731018950688, 3.83529311116720005700264068500, 4.37283475175060562443840853129, 5.08288345322867528329038867526, 6.19744060196011251760500061999, 6.76924117660136728088581483024, 7.976041230983647907367889184985, 8.434730402021773477289920520526

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