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Properties

Label 2-17-17.13-c11-0-5
Degree $2$
Conductor $17$
Sign $-0.999 - 0.0324i$
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  + 58.9i·2-s + (500. + 500. i)3-s − 1.42e3·4-s + (2.48e3 + 2.48e3i)5-s + (−2.95e4 + 2.95e4i)6-s + (−731. + 731. i)7-s + 3.67e4i·8-s + 3.24e5i·9-s + (−1.46e5 + 1.46e5i)10-s + (3.46e5 − 3.46e5i)11-s + (−7.13e5 − 7.13e5i)12-s + 6.89e5·13-s + (−4.30e4 − 4.30e4i)14-s + 2.49e6i·15-s − 5.08e6·16-s + (−3.45e6 − 4.72e6i)17-s + ⋯
L(s)  = 1  + 1.30i·2-s + (1.18 + 1.18i)3-s − 0.695·4-s + (0.355 + 0.355i)5-s + (−1.54 + 1.54i)6-s + (−0.0164 + 0.0164i)7-s + 0.396i·8-s + 1.82i·9-s + (−0.463 + 0.463i)10-s + (0.648 − 0.648i)11-s + (−0.827 − 0.827i)12-s + 0.515·13-s + (−0.0214 − 0.0214i)14-s + 0.846i·15-s − 1.21·16-s + (−0.589 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0324i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.999 - 0.0324i$
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: \((2,\ 17,\ (\ :11/2),\ -0.999 - 0.0324i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0468359 + 2.89019i\)
\(L(\frac12)\) \(\approx\) \(0.0468359 + 2.89019i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.45e6 + 4.72e6i)T \)
good2 \( 1 - 58.9iT - 2.04e3T^{2} \)
3 \( 1 + (-500. - 500. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-2.48e3 - 2.48e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (731. - 731. i)T - 1.97e9iT^{2} \)
11 \( 1 + (-3.46e5 + 3.46e5i)T - 2.85e11iT^{2} \)
13 \( 1 - 6.89e5T + 1.79e12T^{2} \)
19 \( 1 + 6.20e6iT - 1.16e14T^{2} \)
23 \( 1 + (-1.68e7 + 1.68e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (4.06e6 + 4.06e6i)T + 1.22e16iT^{2} \)
31 \( 1 + (-2.23e8 - 2.23e8i)T + 2.54e16iT^{2} \)
37 \( 1 + (3.73e8 + 3.73e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (-2.51e8 + 2.51e8i)T - 5.50e17iT^{2} \)
43 \( 1 - 1.69e9iT - 9.29e17T^{2} \)
47 \( 1 + 2.20e9T + 2.47e18T^{2} \)
53 \( 1 - 3.92e9iT - 9.26e18T^{2} \)
59 \( 1 + 4.81e9iT - 3.01e19T^{2} \)
61 \( 1 + (-1.63e9 + 1.63e9i)T - 4.35e19iT^{2} \)
67 \( 1 - 1.86e10T + 1.22e20T^{2} \)
71 \( 1 + (4.42e9 + 4.42e9i)T + 2.31e20iT^{2} \)
73 \( 1 + (1.34e10 + 1.34e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (-2.24e10 + 2.24e10i)T - 7.47e20iT^{2} \)
83 \( 1 - 3.09e10iT - 1.28e21T^{2} \)
89 \( 1 + 3.26e10T + 2.77e21T^{2} \)
97 \( 1 + (9.24e9 + 9.24e9i)T + 7.15e21iT^{2} \)
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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

−16.25873941205794268123802068784, −15.61598923227709457935988334439, −14.39282778254990794406708647257, −13.83897821398149450033226918591, −10.92814137033349798905746843735, −9.268874356300522360827193160249, −8.342142727338402909204587890384, −6.55367012066782140706262091298, −4.68566117197314983259528527368, −2.84880255663496130906921635326, 1.23260324146340112687844576681, 2.09439948015623290783800389845, 3.66546869660272001378693010450, 6.77903101709172386426636265416, 8.533264502960412605970593722201, 9.784623171029489221500119849453, 11.70350791191310914544303678898, 12.89562838276578682070288608975, 13.56390591811664757273209296325, 15.10648111869556382269968997635

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