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Properties

Label 2-17-17.13-c11-0-11
Degree $2$
Conductor $17$
Sign $0.765 + 0.643i$
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  − 21.1i·2-s + (485. + 485. i)3-s + 1.60e3·4-s + (−8.77e3 − 8.77e3i)5-s + (1.02e4 − 1.02e4i)6-s + (4.30e4 − 4.30e4i)7-s − 7.71e4i·8-s + 2.94e5i·9-s + (−1.85e5 + 1.85e5i)10-s + (9.67e4 − 9.67e4i)11-s + (7.77e5 + 7.77e5i)12-s + 1.42e6·13-s + (−9.10e5 − 9.10e5i)14-s − 8.52e6i·15-s + 1.64e6·16-s + (−2.15e5 + 5.85e6i)17-s + ⋯
L(s)  = 1  − 0.467i·2-s + (1.15 + 1.15i)3-s + 0.781·4-s + (−1.25 − 1.25i)5-s + (0.539 − 0.539i)6-s + (0.968 − 0.968i)7-s − 0.832i·8-s + 1.66i·9-s + (−0.587 + 0.587i)10-s + (0.181 − 0.181i)11-s + (0.901 + 0.901i)12-s + 1.06·13-s + (−0.452 − 0.452i)14-s − 2.89i·15-s + 0.392·16-s + (−0.0367 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.765 + 0.643i$
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: \((2,\ 17,\ (\ :11/2),\ 0.765 + 0.643i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.71639 - 0.991092i\)
\(L(\frac12)\) \(\approx\) \(2.71639 - 0.991092i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.15e5 - 5.85e6i)T \)
good2 \( 1 + 21.1iT - 2.04e3T^{2} \)
3 \( 1 + (-485. - 485. i)T + 1.77e5iT^{2} \)
5 \( 1 + (8.77e3 + 8.77e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (-4.30e4 + 4.30e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (-9.67e4 + 9.67e4i)T - 2.85e11iT^{2} \)
13 \( 1 - 1.42e6T + 1.79e12T^{2} \)
19 \( 1 + 1.08e7iT - 1.16e14T^{2} \)
23 \( 1 + (1.21e7 - 1.21e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (3.88e7 + 3.88e7i)T + 1.22e16iT^{2} \)
31 \( 1 + (-7.83e7 - 7.83e7i)T + 2.54e16iT^{2} \)
37 \( 1 + (-8.70e7 - 8.70e7i)T + 1.77e17iT^{2} \)
41 \( 1 + (4.97e8 - 4.97e8i)T - 5.50e17iT^{2} \)
43 \( 1 + 1.02e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.14e9T + 2.47e18T^{2} \)
53 \( 1 - 1.62e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.07e9iT - 3.01e19T^{2} \)
61 \( 1 + (7.13e9 - 7.13e9i)T - 4.35e19iT^{2} \)
67 \( 1 + 6.49e9T + 1.22e20T^{2} \)
71 \( 1 + (-1.95e10 - 1.95e10i)T + 2.31e20iT^{2} \)
73 \( 1 + (1.92e10 + 1.92e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (-4.88e9 + 4.88e9i)T - 7.47e20iT^{2} \)
83 \( 1 - 1.92e9iT - 1.28e21T^{2} \)
89 \( 1 + 9.65e9T + 2.77e21T^{2} \)
97 \( 1 + (-8.73e10 - 8.73e10i)T + 7.15e21iT^{2} \)
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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

−15.83798993358522267854621989315, −15.08423497512511146010232037300, −13.41128965434419452403685085438, −11.61613977973912146907761796326, −10.56723994799056289781171384095, −8.764217409139608823348571997565, −7.80944972023326414824908138367, −4.42616970375231808715571579655, −3.59221615049328657994740182548, −1.26593293561763519844673102961, 1.93039739253590542425156528697, 3.17527736665818344238489471504, 6.47078831939478576302992748387, 7.66339490038085501113625062828, 8.323179084401673285602423953047, 11.23447988057004109331340656257, 12.09586398903519123261079907056, 14.22558168196726733739651809883, 14.86021263738947862859484322704, 15.82497968939978945637397700023

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