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Properties

Label 2-17-17.8-c5-0-4
Degree $2$
Conductor $17$
Sign $0.989 + 0.143i$
Analytic cond. $2.72652$
Root an. cond. $1.65121$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.77 + 5.77i)2-s + (8.99 − 21.7i)3-s − 34.5i·4-s + (33.8 + 14.0i)5-s + (73.3 + 177. i)6-s + (191. − 79.1i)7-s + (14.9 + 14.9i)8-s + (−218. − 218. i)9-s + (−275. + 114. i)10-s + (34.6 + 83.6i)11-s + (−751. − 311. i)12-s − 737. i·13-s + (−645. + 1.55e3i)14-s + (608. − 608. i)15-s + 934.·16-s + (−259. + 1.16e3i)17-s + ⋯
L(s)  = 1  + (−1.02 + 1.02i)2-s + (0.576 − 1.39i)3-s − 1.08i·4-s + (0.604 + 0.250i)5-s + (0.832 + 2.00i)6-s + (1.47 − 0.610i)7-s + (0.0827 + 0.0827i)8-s + (−0.900 − 0.900i)9-s + (−0.872 + 0.361i)10-s + (0.0863 + 0.208i)11-s + (−1.50 − 0.623i)12-s − 1.21i·13-s + (−0.880 + 2.12i)14-s + (0.698 − 0.698i)15-s + 0.912·16-s + (−0.218 + 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.989 + 0.143i$
Analytic conductor: \(2.72652\)
Root analytic conductor: \(1.65121\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :5/2),\ 0.989 + 0.143i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.11049 - 0.0799952i\)
\(L(\frac12)\) \(\approx\) \(1.11049 - 0.0799952i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (259. - 1.16e3i)T \)
good2 \( 1 + (5.77 - 5.77i)T - 32iT^{2} \)
3 \( 1 + (-8.99 + 21.7i)T + (-171. - 171. i)T^{2} \)
5 \( 1 + (-33.8 - 14.0i)T + (2.20e3 + 2.20e3i)T^{2} \)
7 \( 1 + (-191. + 79.1i)T + (1.18e4 - 1.18e4i)T^{2} \)
11 \( 1 + (-34.6 - 83.6i)T + (-1.13e5 + 1.13e5i)T^{2} \)
13 \( 1 + 737. iT - 3.71e5T^{2} \)
19 \( 1 + (1.94e3 - 1.94e3i)T - 2.47e6iT^{2} \)
23 \( 1 + (-644. - 1.55e3i)T + (-4.55e6 + 4.55e6i)T^{2} \)
29 \( 1 + (-3.16e3 - 1.31e3i)T + (1.45e7 + 1.45e7i)T^{2} \)
31 \( 1 + (-388. + 937. i)T + (-2.02e7 - 2.02e7i)T^{2} \)
37 \( 1 + (1.03e3 - 2.49e3i)T + (-4.90e7 - 4.90e7i)T^{2} \)
41 \( 1 + (7.51e3 - 3.11e3i)T + (8.19e7 - 8.19e7i)T^{2} \)
43 \( 1 + (-8.03e3 - 8.03e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 5.99e3iT - 2.29e8T^{2} \)
53 \( 1 + (1.06e4 - 1.06e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (2.12e4 + 2.12e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-6.24e3 + 2.58e3i)T + (5.97e8 - 5.97e8i)T^{2} \)
67 \( 1 + 5.20e4T + 1.35e9T^{2} \)
71 \( 1 + (-371. + 897. i)T + (-1.27e9 - 1.27e9i)T^{2} \)
73 \( 1 + (-6.49e4 - 2.69e4i)T + (1.46e9 + 1.46e9i)T^{2} \)
79 \( 1 + (-7.86e3 - 1.89e4i)T + (-2.17e9 + 2.17e9i)T^{2} \)
83 \( 1 + (2.40e4 - 2.40e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 7.32e4iT - 5.58e9T^{2} \)
97 \( 1 + (-1.15e5 - 4.76e4i)T + (6.07e9 + 6.07e9i)T^{2} \)
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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

−17.67248420505505268048694884518, −17.25055600505516826800870249300, −15.04136117925271509772695439866, −14.11986122957164463421168794692, −12.65692902495738384805235014848, −10.41821381492927686719392235397, −8.316348679602351932745195987819, −7.74964263283599529887665897867, −6.24696166547699348866924100753, −1.49659663412415665217750265300, 2.23779929152534006323076785207, 4.75944644265413037958512276622, 8.696554442504151942181985669038, 9.211951123772780093592763007552, 10.66665032312658415832860445226, 11.66187196614287892882959212918, 14.09877688172472464414320233434, 15.25936022016840639048622368574, 16.90550735667017527076411302401, 17.97636884085362450956036867658

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