Properties

Label 2-1617-1.1-c1-0-3
Degree $2$
Conductor $1617$
Sign $1$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s − 3-s + 4.85·4-s − 2.23·5-s + 2.61·6-s − 7.47·8-s + 9-s + 5.85·10-s − 11-s − 4.85·12-s + 3.47·13-s + 2.23·15-s + 9.85·16-s + 6·17-s − 2.61·18-s − 4.23·19-s − 10.8·20-s + 2.61·22-s − 2.47·23-s + 7.47·24-s − 9.09·26-s − 27-s − 10.2·29-s − 5.85·30-s + 8.47·31-s − 10.8·32-s + 33-s + ⋯
L(s)  = 1  − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.999·5-s + 1.06·6-s − 2.64·8-s + 0.333·9-s + 1.85·10-s − 0.301·11-s − 1.40·12-s + 0.962·13-s + 0.577·15-s + 2.46·16-s + 1.45·17-s − 0.617·18-s − 0.971·19-s − 2.42·20-s + 0.558·22-s − 0.515·23-s + 1.52·24-s − 1.78·26-s − 0.192·27-s − 1.90·29-s − 1.06·30-s + 1.52·31-s − 1.91·32-s + 0.174·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3705266302\)
\(L(\frac12)\) \(\approx\) \(0.3705266302\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + 2.61T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4.23T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 8.47T + 31T^{2} \)
37 \( 1 + 0.527T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 6.94T + 61T^{2} \)
67 \( 1 - 0.708T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 2.47T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 6.94T + 97T^{2} \)
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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

−9.400317977913097710570593126020, −8.524649432781679522984719247707, −7.79411292934807206931554205359, −7.50606998763773376492684119385, −6.34905405829755616043889290647, −5.77257477189267214362641864231, −4.22868064781601819288147691951, −3.16495715504391030713362267660, −1.74450108306809813732576957408, −0.58023986852057187785248996683, 0.58023986852057187785248996683, 1.74450108306809813732576957408, 3.16495715504391030713362267660, 4.22868064781601819288147691951, 5.77257477189267214362641864231, 6.34905405829755616043889290647, 7.50606998763773376492684119385, 7.79411292934807206931554205359, 8.524649432781679522984719247707, 9.400317977913097710570593126020

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