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Properties

Label 2-17-17.13-c9-0-1
Degree $2$
Conductor $17$
Sign $-0.633 + 0.773i$
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  + 36.7i·2-s + (151. + 151. i)3-s − 839.·4-s + (−1.54e3 − 1.54e3i)5-s + (−5.55e3 + 5.55e3i)6-s + (−1.02e3 + 1.02e3i)7-s − 1.20e4i·8-s + 2.59e4i·9-s + (5.68e4 − 5.68e4i)10-s + (−5.37e4 + 5.37e4i)11-s + (−1.26e5 − 1.26e5i)12-s + 6.18e4·13-s + (−3.75e4 − 3.75e4i)14-s − 4.67e5i·15-s + 1.26e4·16-s + (3.44e5 + 8.10e3i)17-s + ⋯
L(s)  = 1  + 1.62i·2-s + (1.07 + 1.07i)3-s − 1.63·4-s + (−1.10 − 1.10i)5-s + (−1.75 + 1.75i)6-s + (−0.160 + 0.160i)7-s − 1.03i·8-s + 1.32i·9-s + (1.79 − 1.79i)10-s + (−1.10 + 1.10i)11-s + (−1.76 − 1.76i)12-s + 0.600·13-s + (−0.261 − 0.261i)14-s − 2.38i·15-s + 0.0484·16-s + (0.999 + 0.0235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.633 + 0.773i$
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: \((2,\ 17,\ (\ :9/2),\ -0.633 + 0.773i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.587250 - 1.24039i\)
\(L(\frac12)\) \(\approx\) \(0.587250 - 1.24039i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.44e5 - 8.10e3i)T \)
good2 \( 1 - 36.7iT - 512T^{2} \)
3 \( 1 + (-151. - 151. i)T + 1.96e4iT^{2} \)
5 \( 1 + (1.54e3 + 1.54e3i)T + 1.95e6iT^{2} \)
7 \( 1 + (1.02e3 - 1.02e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (5.37e4 - 5.37e4i)T - 2.35e9iT^{2} \)
13 \( 1 - 6.18e4T + 1.06e10T^{2} \)
19 \( 1 - 1.04e6iT - 3.22e11T^{2} \)
23 \( 1 + (-1.90e5 + 1.90e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (1.22e6 + 1.22e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-2.56e6 - 2.56e6i)T + 2.64e13iT^{2} \)
37 \( 1 + (2.54e6 + 2.54e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (1.43e7 - 1.43e7i)T - 3.27e14iT^{2} \)
43 \( 1 + 7.95e6iT - 5.02e14T^{2} \)
47 \( 1 - 4.08e7T + 1.11e15T^{2} \)
53 \( 1 + 3.85e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.05e8iT - 8.66e15T^{2} \)
61 \( 1 + (-1.80e7 + 1.80e7i)T - 1.16e16iT^{2} \)
67 \( 1 - 2.50e8T + 2.72e16T^{2} \)
71 \( 1 + (5.97e7 + 5.97e7i)T + 4.58e16iT^{2} \)
73 \( 1 + (-1.19e8 - 1.19e8i)T + 5.88e16iT^{2} \)
79 \( 1 + (3.85e8 - 3.85e8i)T - 1.19e17iT^{2} \)
83 \( 1 - 4.07e8iT - 1.86e17T^{2} \)
89 \( 1 + 4.72e8T + 3.50e17T^{2} \)
97 \( 1 + (-2.90e8 - 2.90e8i)T + 7.60e17iT^{2} \)
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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

−16.79059832851310215450067046028, −15.86546381127708142459842968085, −15.39607900761065244423207749335, −14.25911647723545042507032650385, −12.58609764826987312559274121410, −9.846614950306912679135449937281, −8.430187392721742618229078658756, −7.82127595382706583978748628004, −5.17852952761424384882900660517, −3.93902875911649449371818104133, 0.61445084414692972378979606829, 2.68445006246085517899485504804, 3.41013807526964341620163791888, 7.27387078856270858250948314630, 8.604153894328976301987192016351, 10.62148381502017131983974586698, 11.63479381363372178879339209812, 13.09874228512551539186496770841, 13.90000647313263823032543903489, 15.50104459310147083022232926906

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