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Properties

Label 2-17-17.13-c9-0-4
Degree $2$
Conductor $17$
Sign $-0.0843 - 0.996i$
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  + 17.1i·2-s + (97.3 + 97.3i)3-s + 217.·4-s + (882. + 882. i)5-s + (−1.67e3 + 1.67e3i)6-s + (7.46e3 − 7.46e3i)7-s + 1.25e4i·8-s − 717. i·9-s + (−1.51e4 + 1.51e4i)10-s + (−3.49e4 + 3.49e4i)11-s + (2.12e4 + 2.12e4i)12-s − 5.43e4·13-s + (1.27e5 + 1.27e5i)14-s + 1.71e5i·15-s − 1.03e5·16-s + (−2.52e5 + 2.34e5i)17-s + ⋯
L(s)  = 1  + 0.757i·2-s + (0.694 + 0.694i)3-s + 0.425·4-s + (0.631 + 0.631i)5-s + (−0.526 + 0.526i)6-s + (1.17 − 1.17i)7-s + 1.08i·8-s − 0.0364i·9-s + (−0.478 + 0.478i)10-s + (−0.719 + 0.719i)11-s + (0.295 + 0.295i)12-s − 0.527·13-s + (0.890 + 0.890i)14-s + 0.876i·15-s − 0.393·16-s + (−0.733 + 0.679i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0843 - 0.996i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0843 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.0843 - 0.996i$
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: \((2,\ 17,\ (\ :9/2),\ -0.0843 - 0.996i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.84847 + 2.01162i\)
\(L(\frac12)\) \(\approx\) \(1.84847 + 2.01162i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.52e5 - 2.34e5i)T \)
good2 \( 1 - 17.1iT - 512T^{2} \)
3 \( 1 + (-97.3 - 97.3i)T + 1.96e4iT^{2} \)
5 \( 1 + (-882. - 882. i)T + 1.95e6iT^{2} \)
7 \( 1 + (-7.46e3 + 7.46e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (3.49e4 - 3.49e4i)T - 2.35e9iT^{2} \)
13 \( 1 + 5.43e4T + 1.06e10T^{2} \)
19 \( 1 + 7.16e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.43e5 + 1.43e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (4.58e6 + 4.58e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-2.48e6 - 2.48e6i)T + 2.64e13iT^{2} \)
37 \( 1 + (-1.31e7 - 1.31e7i)T + 1.29e14iT^{2} \)
41 \( 1 + (1.35e7 - 1.35e7i)T - 3.27e14iT^{2} \)
43 \( 1 + 1.47e7iT - 5.02e14T^{2} \)
47 \( 1 - 7.91e6T + 1.11e15T^{2} \)
53 \( 1 + 1.83e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.19e7iT - 8.66e15T^{2} \)
61 \( 1 + (-1.90e7 + 1.90e7i)T - 1.16e16iT^{2} \)
67 \( 1 + 5.75e7T + 2.72e16T^{2} \)
71 \( 1 + (2.02e7 + 2.02e7i)T + 4.58e16iT^{2} \)
73 \( 1 + (-2.61e8 - 2.61e8i)T + 5.88e16iT^{2} \)
79 \( 1 + (3.71e7 - 3.71e7i)T - 1.19e17iT^{2} \)
83 \( 1 - 1.13e8iT - 1.86e17T^{2} \)
89 \( 1 - 6.81e8T + 3.50e17T^{2} \)
97 \( 1 + (1.14e9 + 1.14e9i)T + 7.60e17iT^{2} \)
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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

−17.11732168457061021853579132692, −15.33657674926864685632329008906, −14.79522322324882622628194757817, −13.66439217470902227652830433517, −11.15623600828124243673878122403, −10.02478796590983307169874029887, −8.087533183446220873486994700255, −6.79560824721363353483036184074, −4.65112120404948410097671734800, −2.34458441198641241682608690319, 1.61046157679809915444452312258, 2.52202994438669414972475114364, 5.45197898986730049258173068982, 7.74901002784697559572255559081, 9.095403639134393868119795454208, 11.00037889283630196350851914750, 12.30930354968412488877800469064, 13.39449376488946081260505916633, 14.83087573111872627680861868659, 16.39371080809645581160718350428

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