Properties

Label 2-1617-1.1-c3-0-84
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  − 0.347·2-s − 3·3-s − 7.87·4-s − 10.0·5-s + 1.04·6-s + 5.52·8-s + 9·9-s + 3.50·10-s + 11·11-s + 23.6·12-s − 57.3·13-s + 30.2·15-s + 61.1·16-s − 75.9·17-s − 3.13·18-s − 28.3·19-s + 79.4·20-s − 3.82·22-s + 4.21·23-s − 16.5·24-s − 23.4·25-s + 19.9·26-s − 27·27-s + 176.·29-s − 10.5·30-s + 214.·31-s − 65.4·32-s + ⋯
L(s)  = 1  − 0.122·2-s − 0.577·3-s − 0.984·4-s − 0.901·5-s + 0.0710·6-s + 0.244·8-s + 0.333·9-s + 0.110·10-s + 0.301·11-s + 0.568·12-s − 1.22·13-s + 0.520·15-s + 0.954·16-s − 1.08·17-s − 0.0409·18-s − 0.341·19-s + 0.887·20-s − 0.0370·22-s + 0.0381·23-s − 0.140·24-s − 0.187·25-s + 0.150·26-s − 0.192·27-s + 1.13·29-s − 0.0640·30-s + 1.24·31-s − 0.361·32-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 \( 1 + 0.347T + 8T^{2} \)
5 \( 1 + 10.0T + 125T^{2} \)
13 \( 1 + 57.3T + 2.19e3T^{2} \)
17 \( 1 + 75.9T + 4.91e3T^{2} \)
19 \( 1 + 28.3T + 6.85e3T^{2} \)
23 \( 1 - 4.21T + 1.21e4T^{2} \)
29 \( 1 - 176.T + 2.43e4T^{2} \)
31 \( 1 - 214.T + 2.97e4T^{2} \)
37 \( 1 - 212.T + 5.06e4T^{2} \)
41 \( 1 + 221.T + 6.89e4T^{2} \)
43 \( 1 + 30.9T + 7.95e4T^{2} \)
47 \( 1 - 570.T + 1.03e5T^{2} \)
53 \( 1 + 278.T + 1.48e5T^{2} \)
59 \( 1 - 129.T + 2.05e5T^{2} \)
61 \( 1 - 496.T + 2.26e5T^{2} \)
67 \( 1 - 507.T + 3.00e5T^{2} \)
71 \( 1 - 291.T + 3.57e5T^{2} \)
73 \( 1 + 376.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 306.T + 5.71e5T^{2} \)
89 \( 1 - 16.7T + 7.04e5T^{2} \)
97 \( 1 - 501.T + 9.12e5T^{2} \)
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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

−8.565723813099495376975827489866, −7.947156854641795158750711915672, −7.07378147265053226108794795447, −6.24708840961776320452068128615, −5.06034212859228713873465300517, −4.50839506060894873923394848350, −3.83035751694783833997821349192, −2.46684148078146555052671272845, −0.853451782916020251623180076413, 0, 0.853451782916020251623180076413, 2.46684148078146555052671272845, 3.83035751694783833997821349192, 4.50839506060894873923394848350, 5.06034212859228713873465300517, 6.24708840961776320452068128615, 7.07378147265053226108794795447, 7.947156854641795158750711915672, 8.565723813099495376975827489866

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