Properties

Label 2-17-17.9-c9-0-9
Degree $2$
Conductor $17$
Sign $-0.757 + 0.653i$
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  + (−2.23 + 2.23i)2-s + (−45.0 − 18.6i)3-s + 502. i·4-s + (265. − 640. i)5-s + (142. − 58.9i)6-s + (−537. − 1.29e3i)7-s + (−2.26e3 − 2.26e3i)8-s + (−1.22e4 − 1.22e4i)9-s + (838. + 2.02e3i)10-s + (−6.54e4 + 2.71e4i)11-s + (9.36e3 − 2.25e4i)12-s − 1.80e5i·13-s + (4.09e3 + 1.69e3i)14-s + (−2.38e4 + 2.38e4i)15-s − 2.46e5·16-s + (4.12e4 − 3.41e5i)17-s + ⋯
L(s)  = 1  + (−0.0987 + 0.0987i)2-s + (−0.320 − 0.132i)3-s + 0.980i·4-s + (0.189 − 0.458i)5-s + (0.0448 − 0.0185i)6-s + (−0.0845 − 0.204i)7-s + (−0.195 − 0.195i)8-s + (−0.621 − 0.621i)9-s + (0.0265 + 0.0639i)10-s + (−1.34 + 0.558i)11-s + (0.130 − 0.314i)12-s − 1.75i·13-s + (0.0285 + 0.0118i)14-s + (−0.121 + 0.121i)15-s − 0.941·16-s + (0.119 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.757 + 0.653i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.757 + 0.653i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.123713 - 0.332707i\)
\(L(\frac12)\) \(\approx\) \(0.123713 - 0.332707i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.12e4 + 3.41e5i)T \)
good2 \( 1 + (2.23 - 2.23i)T - 512iT^{2} \)
3 \( 1 + (45.0 + 18.6i)T + (1.39e4 + 1.39e4i)T^{2} \)
5 \( 1 + (-265. + 640. i)T + (-1.38e6 - 1.38e6i)T^{2} \)
7 \( 1 + (537. + 1.29e3i)T + (-2.85e7 + 2.85e7i)T^{2} \)
11 \( 1 + (6.54e4 - 2.71e4i)T + (1.66e9 - 1.66e9i)T^{2} \)
13 \( 1 + 1.80e5iT - 1.06e10T^{2} \)
19 \( 1 + (5.99e5 - 5.99e5i)T - 3.22e11iT^{2} \)
23 \( 1 + (-1.45e6 + 6.03e5i)T + (1.27e12 - 1.27e12i)T^{2} \)
29 \( 1 + (1.18e6 - 2.86e6i)T + (-1.02e13 - 1.02e13i)T^{2} \)
31 \( 1 + (3.10e6 + 1.28e6i)T + (1.86e13 + 1.86e13i)T^{2} \)
37 \( 1 + (1.75e7 + 7.27e6i)T + (9.18e13 + 9.18e13i)T^{2} \)
41 \( 1 + (-3.30e6 - 7.98e6i)T + (-2.31e14 + 2.31e14i)T^{2} \)
43 \( 1 + (7.27e6 + 7.27e6i)T + 5.02e14iT^{2} \)
47 \( 1 - 5.71e6iT - 1.11e15T^{2} \)
53 \( 1 + (5.32e7 - 5.32e7i)T - 3.29e15iT^{2} \)
59 \( 1 + (-1.00e8 - 1.00e8i)T + 8.66e15iT^{2} \)
61 \( 1 + (-4.03e4 - 9.74e4i)T + (-8.26e15 + 8.26e15i)T^{2} \)
67 \( 1 + 1.57e6T + 2.72e16T^{2} \)
71 \( 1 + (3.00e8 + 1.24e8i)T + (3.24e16 + 3.24e16i)T^{2} \)
73 \( 1 + (7.03e6 - 1.69e7i)T + (-4.16e16 - 4.16e16i)T^{2} \)
79 \( 1 + (-1.35e8 + 5.63e7i)T + (8.47e16 - 8.47e16i)T^{2} \)
83 \( 1 + (-2.62e8 + 2.62e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 3.54e8iT - 3.50e17T^{2} \)
97 \( 1 + (-5.89e8 + 1.42e9i)T + (-5.37e17 - 5.37e17i)T^{2} \)
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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

−16.48001728986359881832365377094, −15.08734932317466057100733510456, −13.03686800746006500626631355412, −12.42773988547731735704293022918, −10.62332480162201495948651070456, −8.736096537068800694951451137418, −7.37906705677569696479528803992, −5.32049165015847802630806432744, −3.05104156539369474832351707188, −0.17606778700386965858400396365, 2.24895818913489888925930773830, 5.08163236425657428090584378628, 6.49685966584458523164411826248, 8.795148805085886710188682999648, 10.50065536384180756817483554377, 11.25551737739491127463788152790, 13.41151261718665122021380225132, 14.58853375613273066259772966230, 15.86956359890694586322282415804, 17.26536101662279209857742112187

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