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  + 5.44·2-s + 106.·3-s − 482.·4-s + 1.30e3·5-s + 579.·6-s + 9.19e3·7-s − 5.41e3·8-s − 8.34e3·9-s + 7.09e3·10-s + 6.22e4·11-s − 5.13e4·12-s + 1.41e5·13-s + 5.00e4·14-s + 1.38e5·15-s + 2.17e5·16-s + 8.35e4·17-s − 4.54e4·18-s − 9.41e5·19-s − 6.28e5·20-s + 9.79e5·21-s + 3.38e5·22-s + 5.68e5·23-s − 5.76e5·24-s − 2.52e5·25-s + 7.72e5·26-s − 2.98e6·27-s − 4.43e6·28-s + ⋯
L(s)  = 1  + 0.240·2-s + 0.758·3-s − 0.942·4-s + 0.933·5-s + 0.182·6-s + 1.44·7-s − 0.467·8-s − 0.424·9-s + 0.224·10-s + 1.28·11-s − 0.714·12-s + 1.37·13-s + 0.348·14-s + 0.708·15-s + 0.829·16-s + 0.242·17-s − 0.102·18-s − 1.65·19-s − 0.878·20-s + 1.09·21-s + 0.308·22-s + 0.423·23-s − 0.354·24-s − 0.129·25-s + 0.331·26-s − 1.08·27-s − 1.36·28-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

\(L(5)\) \(\approx\) \(2.56171\)
\(L(\frac12)\) \(\approx\) \(2.56171\)
\(L(\frac{11}{2})\) 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 \( 1 - 5.44T + 512T^{2} \)
3 \( 1 - 106.T + 1.96e4T^{2} \)
5 \( 1 - 1.30e3T + 1.95e6T^{2} \)
7 \( 1 - 9.19e3T + 4.03e7T^{2} \)
11 \( 1 - 6.22e4T + 2.35e9T^{2} \)
13 \( 1 - 1.41e5T + 1.06e10T^{2} \)
19 \( 1 + 9.41e5T + 3.22e11T^{2} \)
23 \( 1 - 5.68e5T + 1.80e12T^{2} \)
29 \( 1 + 1.83e6T + 1.45e13T^{2} \)
31 \( 1 + 7.34e6T + 2.64e13T^{2} \)
37 \( 1 + 8.01e6T + 1.29e14T^{2} \)
41 \( 1 - 1.95e7T + 3.27e14T^{2} \)
43 \( 1 - 3.46e7T + 5.02e14T^{2} \)
47 \( 1 + 5.63e7T + 1.11e15T^{2} \)
53 \( 1 + 3.28e7T + 3.29e15T^{2} \)
59 \( 1 - 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 5.69e7T + 1.16e16T^{2} \)
67 \( 1 + 1.58e7T + 2.72e16T^{2} \)
71 \( 1 + 9.81e7T + 4.58e16T^{2} \)
73 \( 1 + 6.27e7T + 5.88e16T^{2} \)
79 \( 1 + 1.38e8T + 1.19e17T^{2} \)
83 \( 1 + 6.69e8T + 1.86e17T^{2} \)
89 \( 1 + 4.21e7T + 3.50e17T^{2} \)
97 \( 1 + 4.11e8T + 7.60e17T^{2} \)
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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

−17.14423851373842609617445615111, −14.61290355171031431498985909944, −14.29968062109002246312087683479, −13.09896495790926813976974466517, −11.14362826735001421219031267231, −9.168450703981995425741722503382, −8.403579659933320140942390111264, −5.78859085187022506234082272096, −3.98718451997191803367848058337, −1.64154761238917240061481378123, 1.64154761238917240061481378123, 3.98718451997191803367848058337, 5.78859085187022506234082272096, 8.403579659933320140942390111264, 9.168450703981995425741722503382, 11.14362826735001421219031267231, 13.09896495790926813976974466517, 14.29968062109002246312087683479, 14.61290355171031431498985909944, 17.14423851373842609617445615111

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