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  − 28.6·2-s + 243.·3-s + 308.·4-s + 1.77e3·5-s − 6.98e3·6-s − 9.59e3·7-s + 5.83e3·8-s + 3.98e4·9-s − 5.08e4·10-s + 1.86e4·11-s + 7.52e4·12-s + 1.18e5·13-s + 2.74e5·14-s + 4.33e5·15-s − 3.24e5·16-s + 8.35e4·17-s − 1.14e6·18-s + 3.65e5·19-s + 5.47e5·20-s − 2.34e6·21-s − 5.34e5·22-s + 1.22e6·23-s + 1.42e6·24-s + 1.20e6·25-s − 3.38e6·26-s + 4.91e6·27-s − 2.95e6·28-s + ⋯
L(s)  = 1  − 1.26·2-s + 1.73·3-s + 0.602·4-s + 1.27·5-s − 2.20·6-s − 1.51·7-s + 0.503·8-s + 2.02·9-s − 1.60·10-s + 0.384·11-s + 1.04·12-s + 1.14·13-s + 1.91·14-s + 2.21·15-s − 1.23·16-s + 0.242·17-s − 2.56·18-s + 0.643·19-s + 0.765·20-s − 2.62·21-s − 0.486·22-s + 0.914·23-s + 0.875·24-s + 0.616·25-s − 1.45·26-s + 1.78·27-s − 0.909·28-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

\(L(5)\) \(\approx\) \(1.75431\)
\(L(\frac12)\) \(\approx\) \(1.75431\)
\(L(\frac{11}{2})\) 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 \( 1 + 28.6T + 512T^{2} \)
3 \( 1 - 243.T + 1.96e4T^{2} \)
5 \( 1 - 1.77e3T + 1.95e6T^{2} \)
7 \( 1 + 9.59e3T + 4.03e7T^{2} \)
11 \( 1 - 1.86e4T + 2.35e9T^{2} \)
13 \( 1 - 1.18e5T + 1.06e10T^{2} \)
19 \( 1 - 3.65e5T + 3.22e11T^{2} \)
23 \( 1 - 1.22e6T + 1.80e12T^{2} \)
29 \( 1 + 2.55e6T + 1.45e13T^{2} \)
31 \( 1 - 8.56e6T + 2.64e13T^{2} \)
37 \( 1 + 7.97e6T + 1.29e14T^{2} \)
41 \( 1 + 3.21e7T + 3.27e14T^{2} \)
43 \( 1 + 2.38e7T + 5.02e14T^{2} \)
47 \( 1 + 1.94e7T + 1.11e15T^{2} \)
53 \( 1 + 1.96e7T + 3.29e15T^{2} \)
59 \( 1 - 1.58e7T + 8.66e15T^{2} \)
61 \( 1 + 1.09e8T + 1.16e16T^{2} \)
67 \( 1 + 8.19e7T + 2.72e16T^{2} \)
71 \( 1 - 1.93e8T + 4.58e16T^{2} \)
73 \( 1 + 1.53e7T + 5.88e16T^{2} \)
79 \( 1 - 1.36e8T + 1.19e17T^{2} \)
83 \( 1 + 6.51e8T + 1.86e17T^{2} \)
89 \( 1 + 6.97e8T + 3.50e17T^{2} \)
97 \( 1 + 4.50e8T + 7.60e17T^{2} \)
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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

−16.87123028319226694165067842195, −15.58946119308220525642658762077, −13.77005494354385747018021886422, −13.28769839183506242855959393187, −10.07189144231759195452716004784, −9.458457404612652835252267054814, −8.523490979557632366314934560150, −6.74289187633109225193852640621, −3.19151948276565238362434478355, −1.50121256120600685429285899933, 1.50121256120600685429285899933, 3.19151948276565238362434478355, 6.74289187633109225193852640621, 8.523490979557632366314934560150, 9.458457404612652835252267054814, 10.07189144231759195452716004784, 13.28769839183506242855959393187, 13.77005494354385747018021886422, 15.58946119308220525642658762077, 16.87123028319226694165067842195

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