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  + 43.1·2-s − 171.·3-s + 1.34e3·4-s + 1.53e3·5-s − 7.37e3·6-s + 3.02e3·7-s + 3.60e4·8-s + 9.56e3·9-s + 6.62e4·10-s + 5.19e4·11-s − 2.30e5·12-s − 1.66e5·13-s + 1.30e5·14-s − 2.62e5·15-s + 8.63e5·16-s + 8.35e4·17-s + 4.12e5·18-s − 1.03e6·19-s + 2.06e6·20-s − 5.17e5·21-s + 2.24e6·22-s − 6.47e5·23-s − 6.16e6·24-s + 4.06e5·25-s − 7.17e6·26-s + 1.73e6·27-s + 4.07e6·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 1.21·3-s + 2.63·4-s + 1.09·5-s − 2.32·6-s + 0.476·7-s + 3.10·8-s + 0.486·9-s + 2.09·10-s + 1.07·11-s − 3.20·12-s − 1.61·13-s + 0.908·14-s − 1.33·15-s + 3.29·16-s + 0.242·17-s + 0.926·18-s − 1.82·19-s + 2.89·20-s − 0.581·21-s + 2.03·22-s − 0.482·23-s − 3.79·24-s + 0.208·25-s − 3.07·26-s + 0.626·27-s + 1.25·28-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

\(L(5)\) \(\approx\) \(4.17223\)
\(L(\frac12)\) \(\approx\) \(4.17223\)
\(L(\frac{11}{2})\) 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 \( 1 - 43.1T + 512T^{2} \)
3 \( 1 + 171.T + 1.96e4T^{2} \)
5 \( 1 - 1.53e3T + 1.95e6T^{2} \)
7 \( 1 - 3.02e3T + 4.03e7T^{2} \)
11 \( 1 - 5.19e4T + 2.35e9T^{2} \)
13 \( 1 + 1.66e5T + 1.06e10T^{2} \)
19 \( 1 + 1.03e6T + 3.22e11T^{2} \)
23 \( 1 + 6.47e5T + 1.80e12T^{2} \)
29 \( 1 - 1.01e5T + 1.45e13T^{2} \)
31 \( 1 - 1.03e6T + 2.64e13T^{2} \)
37 \( 1 + 5.58e6T + 1.29e14T^{2} \)
41 \( 1 + 4.18e6T + 3.27e14T^{2} \)
43 \( 1 + 9.60e6T + 5.02e14T^{2} \)
47 \( 1 - 3.98e7T + 1.11e15T^{2} \)
53 \( 1 - 6.22e7T + 3.29e15T^{2} \)
59 \( 1 - 9.60e7T + 8.66e15T^{2} \)
61 \( 1 + 1.86e8T + 1.16e16T^{2} \)
67 \( 1 - 3.73e7T + 2.72e16T^{2} \)
71 \( 1 - 2.04e8T + 4.58e16T^{2} \)
73 \( 1 + 1.95e8T + 5.88e16T^{2} \)
79 \( 1 - 2.72e8T + 1.19e17T^{2} \)
83 \( 1 + 1.96e8T + 1.86e17T^{2} \)
89 \( 1 + 3.93e8T + 3.50e17T^{2} \)
97 \( 1 - 8.75e8T + 7.60e17T^{2} \)
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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

−16.80148426133319821664368038389, −14.92849588036593211185287576832, −14.04680947809212863089612676546, −12.56521244844904325896824088305, −11.72703871362821385519538187084, −10.38762271031204590068908950068, −6.71926124848699301595070906182, −5.71504062958116055510467538005, −4.55855265695563168355632802467, −2.05266453409891427308902586184, 2.05266453409891427308902586184, 4.55855265695563168355632802467, 5.71504062958116055510467538005, 6.71926124848699301595070906182, 10.38762271031204590068908950068, 11.72703871362821385519538187084, 12.56521244844904325896824088305, 14.04680947809212863089612676546, 14.92849588036593211185287576832, 16.80148426133319821664368038389

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