Properties

Label 2-1617-1.1-c3-0-75
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  − 4.31·2-s − 3·3-s + 10.6·4-s − 14.6·5-s + 12.9·6-s − 11.2·8-s + 9·9-s + 63.2·10-s + 11·11-s − 31.8·12-s − 24.6·13-s + 43.9·15-s − 36.2·16-s + 61.9·17-s − 38.8·18-s − 16.6·19-s − 155.·20-s − 47.4·22-s − 50.6·23-s + 33.8·24-s + 89.5·25-s + 106.·26-s − 27·27-s − 100.·29-s − 189.·30-s − 113.·31-s + 246.·32-s + ⋯
L(s)  = 1  − 1.52·2-s − 0.577·3-s + 1.32·4-s − 1.31·5-s + 0.880·6-s − 0.498·8-s + 0.333·9-s + 1.99·10-s + 0.301·11-s − 0.766·12-s − 0.526·13-s + 0.756·15-s − 0.566·16-s + 0.883·17-s − 0.508·18-s − 0.200·19-s − 1.73·20-s − 0.459·22-s − 0.459·23-s + 0.287·24-s + 0.716·25-s + 0.803·26-s − 0.192·27-s − 0.642·29-s − 1.15·30-s − 0.657·31-s + 1.36·32-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 \( 1 + 4.31T + 8T^{2} \)
5 \( 1 + 14.6T + 125T^{2} \)
13 \( 1 + 24.6T + 2.19e3T^{2} \)
17 \( 1 - 61.9T + 4.91e3T^{2} \)
19 \( 1 + 16.6T + 6.85e3T^{2} \)
23 \( 1 + 50.6T + 1.21e4T^{2} \)
29 \( 1 + 100.T + 2.43e4T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 + 1.84T + 5.06e4T^{2} \)
41 \( 1 - 62.0T + 6.89e4T^{2} \)
43 \( 1 + 302.T + 7.95e4T^{2} \)
47 \( 1 + 41.4T + 1.03e5T^{2} \)
53 \( 1 - 678.T + 1.48e5T^{2} \)
59 \( 1 + 144.T + 2.05e5T^{2} \)
61 \( 1 + 228.T + 2.26e5T^{2} \)
67 \( 1 - 482.T + 3.00e5T^{2} \)
71 \( 1 - 559.T + 3.57e5T^{2} \)
73 \( 1 + 218.T + 3.89e5T^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + 187.T + 5.71e5T^{2} \)
89 \( 1 - 34.7T + 7.04e5T^{2} \)
97 \( 1 + 712.T + 9.12e5T^{2} \)
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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

−8.562813775460430908747244377578, −7.85849225350958571845677477397, −7.35507938660292000942841544383, −6.62471830084149068586007297658, −5.44900150457538232813447135093, −4.37586622726986310156378340120, −3.47780765058684485033986910476, −2.00480660665654635277159167754, −0.817282821932240496625609900588, 0, 0.817282821932240496625609900588, 2.00480660665654635277159167754, 3.47780765058684485033986910476, 4.37586622726986310156378340120, 5.44900150457538232813447135093, 6.62471830084149068586007297658, 7.35507938660292000942841544383, 7.85849225350958571845677477397, 8.562813775460430908747244377578

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