Properties

Label 2-1617-1.1-c3-0-24
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.42·2-s + 3·3-s + 11.5·4-s − 2.84·5-s − 13.2·6-s − 15.8·8-s + 9·9-s + 12.6·10-s − 11·11-s + 34.7·12-s − 5.15·13-s − 8.54·15-s − 22.6·16-s − 121.·17-s − 39.8·18-s − 34.8·19-s − 32.9·20-s + 48.6·22-s + 116.·23-s − 47.4·24-s − 116.·25-s + 22.7·26-s + 27·27-s − 69.4·29-s + 37.8·30-s − 140.·31-s + 226.·32-s + ⋯
L(s)  = 1  − 1.56·2-s + 0.577·3-s + 1.44·4-s − 0.254·5-s − 0.903·6-s − 0.699·8-s + 0.333·9-s + 0.398·10-s − 0.301·11-s + 0.835·12-s − 0.109·13-s − 0.147·15-s − 0.353·16-s − 1.73·17-s − 0.521·18-s − 0.420·19-s − 0.368·20-s + 0.471·22-s + 1.05·23-s − 0.403·24-s − 0.935·25-s + 0.171·26-s + 0.192·27-s − 0.444·29-s + 0.230·30-s − 0.814·31-s + 1.25·32-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: \(0\)
Selberg data: \((2,\ 1617,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6764834965\)
\(L(\frac12)\) \(\approx\) \(0.6764834965\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.42T + 8T^{2} \)
5 \( 1 + 2.84T + 125T^{2} \)
13 \( 1 + 5.15T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 + 34.8T + 6.85e3T^{2} \)
23 \( 1 - 116.T + 1.21e4T^{2} \)
29 \( 1 + 69.4T + 2.43e4T^{2} \)
31 \( 1 + 140.T + 2.97e4T^{2} \)
37 \( 1 + 420.T + 5.06e4T^{2} \)
41 \( 1 - 322.T + 6.89e4T^{2} \)
43 \( 1 - 321.T + 7.95e4T^{2} \)
47 \( 1 - 231.T + 1.03e5T^{2} \)
53 \( 1 - 4.91T + 1.48e5T^{2} \)
59 \( 1 + 406.T + 2.05e5T^{2} \)
61 \( 1 - 556.T + 2.26e5T^{2} \)
67 \( 1 - 84.7T + 3.00e5T^{2} \)
71 \( 1 - 49.0T + 3.57e5T^{2} \)
73 \( 1 + 785.T + 3.89e5T^{2} \)
79 \( 1 + 383.T + 4.93e5T^{2} \)
83 \( 1 - 930.T + 5.71e5T^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 - 1.17e3T + 9.12e5T^{2} \)
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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

−8.947355632769483243840650400036, −8.527791580726799853797526746065, −7.48612524589023936467569421851, −7.18021327818016079422293239077, −6.14483887649982209579735570750, −4.80610676343150286493823471873, −3.82432704759850637941606073354, −2.50932926250790549815058638282, −1.81995318623054412485250398184, −0.47370988602242570374313802893, 0.47370988602242570374313802893, 1.81995318623054412485250398184, 2.50932926250790549815058638282, 3.82432704759850637941606073354, 4.80610676343150286493823471873, 6.14483887649982209579735570750, 7.18021327818016079422293239077, 7.48612524589023936467569421851, 8.527791580726799853797526746065, 8.947355632769483243840650400036

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