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Properties

Label 2-17-17.13-c9-0-5
Degree $2$
Conductor $17$
Sign $0.978 + 0.207i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 38.1i·2-s + (−95.6 − 95.6i)3-s − 944.·4-s + (−23.8 − 23.8i)5-s + (3.65e3 − 3.65e3i)6-s + (3.40e3 − 3.40e3i)7-s − 1.65e4i·8-s − 1.37e3i·9-s + (909. − 909. i)10-s + (−3.85e3 + 3.85e3i)11-s + (9.04e4 + 9.04e4i)12-s + 8.45e4·13-s + (1.29e5 + 1.29e5i)14-s + 4.55e3i·15-s + 1.46e5·16-s + (−1.50e5 − 3.09e5i)17-s + ⋯
L(s)  = 1  + 1.68i·2-s + (−0.681 − 0.681i)3-s − 1.84·4-s + (−0.0170 − 0.0170i)5-s + (1.15 − 1.15i)6-s + (0.536 − 0.536i)7-s − 1.42i·8-s − 0.0699i·9-s + (0.0287 − 0.0287i)10-s + (−0.0793 + 0.0793i)11-s + (1.25 + 1.25i)12-s + 0.821·13-s + (0.904 + 0.904i)14-s + 0.0232i·15-s + 0.560·16-s + (−0.438 − 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.939266 - 0.0987373i\)
\(L(\frac12)\) \(\approx\) \(0.939266 - 0.0987373i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.50e5 + 3.09e5i)T \)
good2 \( 1 - 38.1iT - 512T^{2} \)
3 \( 1 + (95.6 + 95.6i)T + 1.96e4iT^{2} \)
5 \( 1 + (23.8 + 23.8i)T + 1.95e6iT^{2} \)
7 \( 1 + (-3.40e3 + 3.40e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (3.85e3 - 3.85e3i)T - 2.35e9iT^{2} \)
13 \( 1 - 8.45e4T + 1.06e10T^{2} \)
19 \( 1 + 7.31e5iT - 3.22e11T^{2} \)
23 \( 1 + (-8.57e5 + 8.57e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (-3.25e6 - 3.25e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (5.70e6 + 5.70e6i)T + 2.64e13iT^{2} \)
37 \( 1 + (6.44e6 + 6.44e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (1.74e7 - 1.74e7i)T - 3.27e14iT^{2} \)
43 \( 1 - 1.54e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.04e7T + 1.11e15T^{2} \)
53 \( 1 + 2.14e5iT - 3.29e15T^{2} \)
59 \( 1 + 1.32e8iT - 8.66e15T^{2} \)
61 \( 1 + (4.51e7 - 4.51e7i)T - 1.16e16iT^{2} \)
67 \( 1 + 4.10e6T + 2.72e16T^{2} \)
71 \( 1 + (-9.21e7 - 9.21e7i)T + 4.58e16iT^{2} \)
73 \( 1 + (8.35e7 + 8.35e7i)T + 5.88e16iT^{2} \)
79 \( 1 + (-1.97e8 + 1.97e8i)T - 1.19e17iT^{2} \)
83 \( 1 + 4.27e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.00e8T + 3.50e17T^{2} \)
97 \( 1 + (7.92e8 + 7.92e8i)T + 7.60e17iT^{2} \)
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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

−16.72191026030510827150169444460, −15.56107892402711394771975500603, −14.20877489887270722048286799067, −13.05243766511914672257365988037, −11.22619637568744993053548767534, −8.873583818499447942952041671502, −7.33315194977624646439921215045, −6.37496495904557009213784521743, −4.79983903915487808034170975847, −0.54302074007206039399461533868, 1.67442194652201133986155890494, 3.76696916657282237257509529828, 5.36409234191138044306684704765, 8.691183338093926428678970911794, 10.30095079127147871875245988773, 11.10554913957941738064326993234, 12.18389858227757778386139009719, 13.63093030590261947690324311337, 15.46227189863564922018144065122, 17.09342244129047230179821525621

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