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  − 16.8·2-s − 116.·3-s − 229.·4-s − 1.10e3·5-s + 1.96e3·6-s − 5.16e3·7-s + 1.24e4·8-s − 6.02e3·9-s + 1.85e4·10-s + 4.45e4·11-s + 2.68e4·12-s + 6.96e4·13-s + 8.68e4·14-s + 1.28e5·15-s − 9.20e4·16-s + 8.35e4·17-s + 1.01e5·18-s − 1.70e5·19-s + 2.53e5·20-s + 6.03e5·21-s − 7.48e5·22-s − 2.00e6·23-s − 1.45e6·24-s − 7.35e5·25-s − 1.17e6·26-s + 3.00e6·27-s + 1.18e6·28-s + ⋯
L(s)  = 1  − 0.742·2-s − 0.833·3-s − 0.447·4-s − 0.789·5-s + 0.619·6-s − 0.812·7-s + 1.07·8-s − 0.305·9-s + 0.586·10-s + 0.917·11-s + 0.373·12-s + 0.676·13-s + 0.604·14-s + 0.657·15-s − 0.351·16-s + 0.242·17-s + 0.227·18-s − 0.299·19-s + 0.353·20-s + 0.677·21-s − 0.681·22-s − 1.49·23-s − 0.896·24-s − 0.376·25-s − 0.502·26-s + 1.08·27-s + 0.364·28-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

\(L(5)\) \(\approx\) \(0.423081\)
\(L(\frac12)\) \(\approx\) \(0.423081\)
\(L(\frac{11}{2})\) 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 \( 1 + 16.8T + 512T^{2} \)
3 \( 1 + 116.T + 1.96e4T^{2} \)
5 \( 1 + 1.10e3T + 1.95e6T^{2} \)
7 \( 1 + 5.16e3T + 4.03e7T^{2} \)
11 \( 1 - 4.45e4T + 2.35e9T^{2} \)
13 \( 1 - 6.96e4T + 1.06e10T^{2} \)
19 \( 1 + 1.70e5T + 3.22e11T^{2} \)
23 \( 1 + 2.00e6T + 1.80e12T^{2} \)
29 \( 1 - 1.55e5T + 1.45e13T^{2} \)
31 \( 1 - 4.27e6T + 2.64e13T^{2} \)
37 \( 1 - 1.51e7T + 1.29e14T^{2} \)
41 \( 1 - 1.59e7T + 3.27e14T^{2} \)
43 \( 1 - 1.49e7T + 5.02e14T^{2} \)
47 \( 1 - 3.36e7T + 1.11e15T^{2} \)
53 \( 1 + 5.50e7T + 3.29e15T^{2} \)
59 \( 1 + 7.94e7T + 8.66e15T^{2} \)
61 \( 1 - 1.27e7T + 1.16e16T^{2} \)
67 \( 1 - 2.73e8T + 2.72e16T^{2} \)
71 \( 1 - 3.88e6T + 4.58e16T^{2} \)
73 \( 1 - 2.32e8T + 5.88e16T^{2} \)
79 \( 1 - 3.38e8T + 1.19e17T^{2} \)
83 \( 1 - 5.74e8T + 1.86e17T^{2} \)
89 \( 1 + 9.82e8T + 3.50e17T^{2} \)
97 \( 1 + 1.03e9T + 7.60e17T^{2} \)
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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

−16.85937991517856665729287539245, −15.93788502167115774254983376808, −14.01540939448793619274921657167, −12.34076933170699821326559890104, −11.07527062996211902781796250507, −9.569886912137996803937703206619, −8.099615789853963887761836891481, −6.19883438744100674546912363288, −4.05334358887555833762083259030, −0.62618691271467735594292864210, 0.62618691271467735594292864210, 4.05334358887555833762083259030, 6.19883438744100674546912363288, 8.099615789853963887761836891481, 9.569886912137996803937703206619, 11.07527062996211902781796250507, 12.34076933170699821326559890104, 14.01540939448793619274921657167, 15.93788502167115774254983376808, 16.85937991517856665729287539245

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