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Properties

Label 2-17-17.13-c11-0-9
Degree $2$
Conductor $17$
Sign $0.853 + 0.520i$
Analytic cond. $13.0618$
Root an. cond. $3.61411$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 85.4i·2-s + (62.9 + 62.9i)3-s − 5.26e3·4-s + (−835. − 835. i)5-s + (−5.37e3 + 5.37e3i)6-s + (−4.91e3 + 4.91e3i)7-s − 2.74e5i·8-s − 1.69e5i·9-s + (7.14e4 − 7.14e4i)10-s + (−1.29e5 + 1.29e5i)11-s + (−3.30e5 − 3.30e5i)12-s − 6.23e5·13-s + (−4.19e5 − 4.19e5i)14-s − 1.05e5i·15-s + 1.27e7·16-s + (6.73e5 + 5.81e6i)17-s + ⋯
L(s)  = 1  + 1.88i·2-s + (0.149 + 0.149i)3-s − 2.56·4-s + (−0.119 − 0.119i)5-s + (−0.282 + 0.282i)6-s + (−0.110 + 0.110i)7-s − 2.96i·8-s − 0.955i·9-s + (0.225 − 0.225i)10-s + (−0.242 + 0.242i)11-s + (−0.383 − 0.383i)12-s − 0.465·13-s + (−0.208 − 0.208i)14-s − 0.0357i·15-s + 3.02·16-s + (0.114 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.853 + 0.520i$
Analytic conductor: \(13.0618\)
Root analytic conductor: \(3.61411\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :11/2),\ 0.853 + 0.520i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.226761 - 0.0636942i\)
\(L(\frac12)\) \(\approx\) \(0.226761 - 0.0636942i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-6.73e5 - 5.81e6i)T \)
good2 \( 1 - 85.4iT - 2.04e3T^{2} \)
3 \( 1 + (-62.9 - 62.9i)T + 1.77e5iT^{2} \)
5 \( 1 + (835. + 835. i)T + 4.88e7iT^{2} \)
7 \( 1 + (4.91e3 - 4.91e3i)T - 1.97e9iT^{2} \)
11 \( 1 + (1.29e5 - 1.29e5i)T - 2.85e11iT^{2} \)
13 \( 1 + 6.23e5T + 1.79e12T^{2} \)
19 \( 1 + 1.24e7iT - 1.16e14T^{2} \)
23 \( 1 + (2.96e7 - 2.96e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (8.67e7 + 8.67e7i)T + 1.22e16iT^{2} \)
31 \( 1 + (-6.08e7 - 6.08e7i)T + 2.54e16iT^{2} \)
37 \( 1 + (5.59e8 + 5.59e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (-1.87e8 + 1.87e8i)T - 5.50e17iT^{2} \)
43 \( 1 + 1.13e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.82e9T + 2.47e18T^{2} \)
53 \( 1 + 1.90e9iT - 9.26e18T^{2} \)
59 \( 1 - 6.28e9iT - 3.01e19T^{2} \)
61 \( 1 + (4.92e9 - 4.92e9i)T - 4.35e19iT^{2} \)
67 \( 1 + 7.40e9T + 1.22e20T^{2} \)
71 \( 1 + (-7.72e9 - 7.72e9i)T + 2.31e20iT^{2} \)
73 \( 1 + (-2.20e10 - 2.20e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (2.78e10 - 2.78e10i)T - 7.47e20iT^{2} \)
83 \( 1 + 2.15e9iT - 1.28e21T^{2} \)
89 \( 1 + 4.31e10T + 2.77e21T^{2} \)
97 \( 1 + (1.05e11 + 1.05e11i)T + 7.15e21iT^{2} \)
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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

−15.81178620577912490870796967460, −15.13522924370751203135540912307, −13.93892820190664569685435559213, −12.49875440790906752116013533197, −9.734592753908505797388395906834, −8.524636616092863207882573323716, −7.10878348503652393280461081502, −5.74352064700551454988410538665, −4.09219961936785840805865671500, −0.098611085427937410191051273319, 1.76708974688525916254162599141, 3.19000566768312858969349894842, 4.92188656861519529387427361376, 8.081971173776212099119464262245, 9.736319938883100835079424954869, 10.83482736152104351356462177230, 12.06408183288094347715551057266, 13.27956125415096579776592763549, 14.30683205009067820314607895572, 16.64886151064349650286678746486

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