Properties

Degree $2$
Conductor $17$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 34.1·2-s + 169.·3-s + 654.·4-s + 195.·5-s + 5.79e3·6-s − 356.·7-s + 4.86e3·8-s + 9.14e3·9-s + 6.66e3·10-s − 2.14e4·11-s + 1.11e5·12-s − 6.20e3·13-s − 1.21e4·14-s + 3.31e4·15-s − 1.68e5·16-s + 8.35e4·17-s + 3.12e5·18-s + 9.07e5·19-s + 1.27e5·20-s − 6.05e4·21-s − 7.33e5·22-s − 1.23e6·23-s + 8.26e5·24-s − 1.91e6·25-s − 2.11e5·26-s − 1.78e6·27-s − 2.33e5·28-s + ⋯
L(s)  = 1  + 1.50·2-s + 1.21·3-s + 1.27·4-s + 0.139·5-s + 1.82·6-s − 0.0561·7-s + 0.419·8-s + 0.464·9-s + 0.210·10-s − 0.442·11-s + 1.54·12-s − 0.0602·13-s − 0.0847·14-s + 0.169·15-s − 0.644·16-s + 0.242·17-s + 0.701·18-s + 1.59·19-s + 0.178·20-s − 0.0679·21-s − 0.667·22-s − 0.920·23-s + 0.508·24-s − 0.980·25-s − 0.0909·26-s − 0.647·27-s − 0.0717·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Motivic weight: \(9\)
Character: $\chi_{17} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(4.84893\)
\(L(\frac12)\) \(\approx\) \(4.84893\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - 8.35e4T \)
good2 \( 1 - 34.1T + 512T^{2} \)
3 \( 1 - 169.T + 1.96e4T^{2} \)
5 \( 1 - 195.T + 1.95e6T^{2} \)
7 \( 1 + 356.T + 4.03e7T^{2} \)
11 \( 1 + 2.14e4T + 2.35e9T^{2} \)
13 \( 1 + 6.20e3T + 1.06e10T^{2} \)
19 \( 1 - 9.07e5T + 3.22e11T^{2} \)
23 \( 1 + 1.23e6T + 1.80e12T^{2} \)
29 \( 1 + 3.01e6T + 1.45e13T^{2} \)
31 \( 1 + 3.34e5T + 2.64e13T^{2} \)
37 \( 1 - 2.06e7T + 1.29e14T^{2} \)
41 \( 1 - 1.47e7T + 3.27e14T^{2} \)
43 \( 1 + 7.75e6T + 5.02e14T^{2} \)
47 \( 1 - 3.19e7T + 1.11e15T^{2} \)
53 \( 1 - 9.46e7T + 3.29e15T^{2} \)
59 \( 1 - 6.01e7T + 8.66e15T^{2} \)
61 \( 1 - 6.05e7T + 1.16e16T^{2} \)
67 \( 1 + 1.26e8T + 2.72e16T^{2} \)
71 \( 1 - 3.33e7T + 4.58e16T^{2} \)
73 \( 1 - 2.85e8T + 5.88e16T^{2} \)
79 \( 1 + 7.60e7T + 1.19e17T^{2} \)
83 \( 1 + 1.73e7T + 1.86e17T^{2} \)
89 \( 1 - 3.96e8T + 3.50e17T^{2} \)
97 \( 1 - 1.13e9T + 7.60e17T^{2} \)
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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

−16.05815647033608435464309266753, −14.91187917938326251278183968247, −13.97754884173371745027133660661, −13.17872465381020833699989618773, −11.69203501931188307819605370270, −9.527000203505408817271030131652, −7.70572490911238671485960578545, −5.64638718591517569820758078715, −3.79306451037682429053245026325, −2.48781855586923147381985174343, 2.48781855586923147381985174343, 3.79306451037682429053245026325, 5.64638718591517569820758078715, 7.70572490911238671485960578545, 9.527000203505408817271030131652, 11.69203501931188307819605370270, 13.17872465381020833699989618773, 13.97754884173371745027133660661, 14.91187917938326251278183968247, 16.05815647033608435464309266753

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