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Properties

Label 2-17-17.13-c15-0-1
Degree $2$
Conductor $17$
Sign $0.984 - 0.177i$
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  − 138. i·2-s + (−4.84e3 − 4.84e3i)3-s + 1.35e4·4-s + (−1.00e5 − 1.00e5i)5-s + (−6.72e5 + 6.72e5i)6-s + (−4.78e5 + 4.78e5i)7-s − 6.42e6i·8-s + 3.26e7i·9-s + (−1.39e7 + 1.39e7i)10-s + (−4.83e7 + 4.83e7i)11-s + (−6.55e7 − 6.55e7i)12-s − 1.83e8·13-s + (6.63e7 + 6.63e7i)14-s + 9.74e8i·15-s − 4.47e8·16-s + (1.26e9 + 1.12e9i)17-s + ⋯
L(s)  = 1  − 0.766i·2-s + (−1.27 − 1.27i)3-s + 0.412·4-s + (−0.575 − 0.575i)5-s + (−0.980 + 0.980i)6-s + (−0.219 + 0.219i)7-s − 1.08i·8-s + 2.27i·9-s + (−0.440 + 0.440i)10-s + (−0.747 + 0.747i)11-s + (−0.528 − 0.528i)12-s − 0.812·13-s + (0.168 + 0.168i)14-s + 1.47i·15-s − 0.416·16-s + (0.745 + 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.984 - 0.177i$
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: \((2,\ 17,\ (\ :15/2),\ 0.984 - 0.177i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.2024688577\)
\(L(\frac12)\) \(\approx\) \(0.2024688577\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.26e9 - 1.12e9i)T \)
good2 \( 1 + 138. iT - 3.27e4T^{2} \)
3 \( 1 + (4.84e3 + 4.84e3i)T + 1.43e7iT^{2} \)
5 \( 1 + (1.00e5 + 1.00e5i)T + 3.05e10iT^{2} \)
7 \( 1 + (4.78e5 - 4.78e5i)T - 4.74e12iT^{2} \)
11 \( 1 + (4.83e7 - 4.83e7i)T - 4.17e15iT^{2} \)
13 \( 1 + 1.83e8T + 5.11e16T^{2} \)
19 \( 1 + 5.39e9iT - 1.51e19T^{2} \)
23 \( 1 + (7.30e8 - 7.30e8i)T - 2.66e20iT^{2} \)
29 \( 1 + (3.26e10 + 3.26e10i)T + 8.62e21iT^{2} \)
31 \( 1 + (-1.49e11 - 1.49e11i)T + 2.34e22iT^{2} \)
37 \( 1 + (-1.11e11 - 1.11e11i)T + 3.33e23iT^{2} \)
41 \( 1 + (5.91e11 - 5.91e11i)T - 1.55e24iT^{2} \)
43 \( 1 + 9.54e11iT - 3.17e24T^{2} \)
47 \( 1 + 3.79e12T + 1.20e25T^{2} \)
53 \( 1 + 1.15e13iT - 7.31e25T^{2} \)
59 \( 1 - 1.64e13iT - 3.65e26T^{2} \)
61 \( 1 + (2.99e13 - 2.99e13i)T - 6.02e26iT^{2} \)
67 \( 1 - 7.81e13T + 2.46e27T^{2} \)
71 \( 1 + (8.25e13 + 8.25e13i)T + 5.87e27iT^{2} \)
73 \( 1 + (-1.18e14 - 1.18e14i)T + 8.90e27iT^{2} \)
79 \( 1 + (7.84e13 - 7.84e13i)T - 2.91e28iT^{2} \)
83 \( 1 - 2.98e14iT - 6.11e28T^{2} \)
89 \( 1 - 5.67e14T + 1.74e29T^{2} \)
97 \( 1 + (-2.52e14 - 2.52e14i)T + 6.33e29iT^{2} \)
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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

−15.57592148970470886835121436698, −13.03955309244356866001596298554, −12.34633583560989764931534117079, −11.60167324376524712625488734632, −10.26521646959238827600167286119, −7.73773715470836206899788639586, −6.60737400004065936410631219334, −4.95099102600415709750955654996, −2.41825698442799875568573162560, −1.05363436168723721920456886730, 0.092969000384985316797361619169, 3.33602071849932488417995568459, 5.10217769907710211730332487496, 6.18077910259968103213300943391, 7.67911013294278855942840538849, 9.978187349097980351978260499527, 11.02797542964615764099960034165, 11.96267259312421046321897797027, 14.55870563007529880768970710395, 15.55660359498132098577668200812

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