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Properties

Label 2-17-17.13-c15-0-0
Degree $2$
Conductor $17$
Sign $-0.915 - 0.403i$
Analytic cond. $24.2578$
Root an. cond. $4.92523$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 119. i·2-s + (4.42e3 + 4.42e3i)3-s + 1.84e4·4-s + (−2.04e5 − 2.04e5i)5-s + (5.28e5 − 5.28e5i)6-s + (−2.13e6 + 2.13e6i)7-s − 6.12e6i·8-s + 2.47e7i·9-s + (−2.44e7 + 2.44e7i)10-s + (−2.88e7 + 2.88e7i)11-s + (8.17e7 + 8.17e7i)12-s − 3.44e8·13-s + (2.55e8 + 2.55e8i)14-s − 1.80e9i·15-s − 1.26e8·16-s + (−4.15e8 − 1.64e9i)17-s + ⋯
L(s)  = 1  − 0.660i·2-s + (1.16 + 1.16i)3-s + 0.564·4-s + (−1.16 − 1.16i)5-s + (0.770 − 0.770i)6-s + (−0.981 + 0.981i)7-s − 1.03i·8-s + 1.72i·9-s + (−0.771 + 0.771i)10-s + (−0.446 + 0.446i)11-s + (0.658 + 0.658i)12-s − 1.52·13-s + (0.647 + 0.647i)14-s − 2.72i·15-s − 0.117·16-s + (−0.245 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.915 - 0.403i$
Analytic conductor: \(24.2578\)
Root analytic conductor: \(4.92523\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :15/2),\ -0.915 - 0.403i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.3529257151\)
\(L(\frac12)\) \(\approx\) \(0.3529257151\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4.15e8 + 1.64e9i)T \)
good2 \( 1 + 119. iT - 3.27e4T^{2} \)
3 \( 1 + (-4.42e3 - 4.42e3i)T + 1.43e7iT^{2} \)
5 \( 1 + (2.04e5 + 2.04e5i)T + 3.05e10iT^{2} \)
7 \( 1 + (2.13e6 - 2.13e6i)T - 4.74e12iT^{2} \)
11 \( 1 + (2.88e7 - 2.88e7i)T - 4.17e15iT^{2} \)
13 \( 1 + 3.44e8T + 5.11e16T^{2} \)
19 \( 1 - 3.48e9iT - 1.51e19T^{2} \)
23 \( 1 + (2.71e9 - 2.71e9i)T - 2.66e20iT^{2} \)
29 \( 1 + (1.94e10 + 1.94e10i)T + 8.62e21iT^{2} \)
31 \( 1 + (-3.93e10 - 3.93e10i)T + 2.34e22iT^{2} \)
37 \( 1 + (7.05e11 + 7.05e11i)T + 3.33e23iT^{2} \)
41 \( 1 + (4.90e11 - 4.90e11i)T - 1.55e24iT^{2} \)
43 \( 1 - 3.27e12iT - 3.17e24T^{2} \)
47 \( 1 + 7.20e10T + 1.20e25T^{2} \)
53 \( 1 + 2.79e12iT - 7.31e25T^{2} \)
59 \( 1 - 9.86e12iT - 3.65e26T^{2} \)
61 \( 1 + (-1.59e13 + 1.59e13i)T - 6.02e26iT^{2} \)
67 \( 1 + 2.49e13T + 2.46e27T^{2} \)
71 \( 1 + (-2.65e13 - 2.65e13i)T + 5.87e27iT^{2} \)
73 \( 1 + (3.91e13 + 3.91e13i)T + 8.90e27iT^{2} \)
79 \( 1 + (-1.45e14 + 1.45e14i)T - 2.91e28iT^{2} \)
83 \( 1 + 1.18e14iT - 6.11e28T^{2} \)
89 \( 1 - 2.23e14T + 1.74e29T^{2} \)
97 \( 1 + (-2.02e14 - 2.02e14i)T + 6.33e29iT^{2} \)
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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

−15.84736094269006804754451202537, −14.91683443036419105295839562670, −12.72672136940486984049299548851, −11.90338083451615454745650883639, −9.973857110592069617560566426729, −9.147588422646541090538600615959, −7.67813645986918135472173619327, −4.83368353901112649993586594625, −3.46822468504802459043863636758, −2.38883372401661644607937032163, 0.090105799810759823485111898814, 2.43739557287814209928027340391, 3.41326352769582807324300383858, 6.82456724506553740703290402093, 7.15027854462331936599302430416, 8.199903391523879180309320197588, 10.53163658522781668286792123607, 12.14474342136020822777614679113, 13.62743006192763472512466055816, 14.75655460779371100988168195030

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