Properties

Label 2-1617-1.1-c1-0-44
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  + 2.59·2-s + 3-s + 4.73·4-s − 2.19·5-s + 2.59·6-s + 7.08·8-s + 9-s − 5.68·10-s + 11-s + 4.73·12-s + 2.95·13-s − 2.19·15-s + 8.92·16-s − 2.59·17-s + 2.59·18-s + 2.35·19-s − 10.3·20-s + 2.59·22-s + 8.92·23-s + 7.08·24-s − 0.195·25-s + 7.66·26-s + 27-s + 4.48·29-s − 5.68·30-s + 1.73·31-s + 8.97·32-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.577·3-s + 2.36·4-s − 0.980·5-s + 1.05·6-s + 2.50·8-s + 0.333·9-s − 1.79·10-s + 0.301·11-s + 1.36·12-s + 0.818·13-s − 0.565·15-s + 2.23·16-s − 0.629·17-s + 0.611·18-s + 0.541·19-s − 2.31·20-s + 0.553·22-s + 1.86·23-s + 1.44·24-s − 0.0390·25-s + 1.50·26-s + 0.192·27-s + 0.833·29-s − 1.03·30-s + 0.310·31-s + 1.58·32-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

\(L(1)\) \(\approx\) \(5.850387132\)
\(L(\frac12)\) \(\approx\) \(5.850387132\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.59T + 2T^{2} \)
5 \( 1 + 2.19T + 5T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 - 8.92T + 23T^{2} \)
29 \( 1 - 4.48T + 29T^{2} \)
31 \( 1 - 1.73T + 31T^{2} \)
37 \( 1 + 4.38T + 37T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 9.17T + 53T^{2} \)
59 \( 1 + 8.92T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 3.25T + 67T^{2} \)
71 \( 1 - 0.994T + 71T^{2} \)
73 \( 1 + 0.294T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 6.26T + 83T^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + 7.51T + 97T^{2} \)
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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

−9.303145430450317948720127972342, −8.386152962411807780490206864923, −7.54285139472778589510240412175, −6.80810490827241025163671797602, −6.11650173624518827151780511815, −4.88548578424811689405684226094, −4.41476566066752217458977125454, −3.37569327522352048045140246715, −3.02373238205682453844855646838, −1.53698908240319652471762740865, 1.53698908240319652471762740865, 3.02373238205682453844855646838, 3.37569327522352048045140246715, 4.41476566066752217458977125454, 4.88548578424811689405684226094, 6.11650173624518827151780511815, 6.80810490827241025163671797602, 7.54285139472778589510240412175, 8.386152962411807780490206864923, 9.303145430450317948720127972342

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