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Properties

Label 2-17-17.13-c9-0-8
Degree $2$
Conductor $17$
Sign $0.665 + 0.746i$
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  + 9.57i·2-s + (−2.66 − 2.66i)3-s + 420.·4-s + (−1.32e3 − 1.32e3i)5-s + (25.4 − 25.4i)6-s + (1.09e3 − 1.09e3i)7-s + 8.92e3i·8-s − 1.96e4i·9-s + (1.26e4 − 1.26e4i)10-s + (4.86e4 − 4.86e4i)11-s + (−1.11e3 − 1.11e3i)12-s + 5.31e4·13-s + (1.05e4 + 1.05e4i)14-s + 7.04e3i·15-s + 1.29e5·16-s + (6.14e4 − 3.38e5i)17-s + ⋯
L(s)  = 1  + 0.423i·2-s + (−0.0189 − 0.0189i)3-s + 0.820·4-s + (−0.947 − 0.947i)5-s + (0.00802 − 0.00802i)6-s + (0.172 − 0.172i)7-s + 0.770i·8-s − 0.999i·9-s + (0.401 − 0.401i)10-s + (1.00 − 1.00i)11-s + (−0.0155 − 0.0155i)12-s + 0.515·13-s + (0.0731 + 0.0731i)14-s + 0.0359i·15-s + 0.494·16-s + (0.178 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.665 + 0.746i$
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: \((2,\ 17,\ (\ :9/2),\ 0.665 + 0.746i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.58849 - 0.711565i\)
\(L(\frac12)\) \(\approx\) \(1.58849 - 0.711565i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-6.14e4 + 3.38e5i)T \)
good2 \( 1 - 9.57iT - 512T^{2} \)
3 \( 1 + (2.66 + 2.66i)T + 1.96e4iT^{2} \)
5 \( 1 + (1.32e3 + 1.32e3i)T + 1.95e6iT^{2} \)
7 \( 1 + (-1.09e3 + 1.09e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (-4.86e4 + 4.86e4i)T - 2.35e9iT^{2} \)
13 \( 1 - 5.31e4T + 1.06e10T^{2} \)
19 \( 1 + 6.43e4iT - 3.22e11T^{2} \)
23 \( 1 + (1.71e6 - 1.71e6i)T - 1.80e12iT^{2} \)
29 \( 1 + (3.30e6 + 3.30e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-2.93e6 - 2.93e6i)T + 2.64e13iT^{2} \)
37 \( 1 + (6.90e6 + 6.90e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (-2.18e7 + 2.18e7i)T - 3.27e14iT^{2} \)
43 \( 1 - 2.81e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.87e7T + 1.11e15T^{2} \)
53 \( 1 + 4.47e6iT - 3.29e15T^{2} \)
59 \( 1 - 1.26e8iT - 8.66e15T^{2} \)
61 \( 1 + (2.41e7 - 2.41e7i)T - 1.16e16iT^{2} \)
67 \( 1 - 1.37e8T + 2.72e16T^{2} \)
71 \( 1 + (2.87e7 + 2.87e7i)T + 4.58e16iT^{2} \)
73 \( 1 + (-1.73e8 - 1.73e8i)T + 5.88e16iT^{2} \)
79 \( 1 + (-2.79e8 + 2.79e8i)T - 1.19e17iT^{2} \)
83 \( 1 + 3.15e8iT - 1.86e17T^{2} \)
89 \( 1 - 4.99e8T + 3.50e17T^{2} \)
97 \( 1 + (-5.54e8 - 5.54e8i)T + 7.60e17iT^{2} \)
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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

−16.27879434414308604970310490769, −15.60630993211282171226530443189, −14.08823246124521733071652596453, −12.08532150773227839812380652483, −11.40880956862425465613138876529, −9.028602457290729563924488060617, −7.63703593181994719852941270023, −5.98125380949640284266360138430, −3.76600652668447157902086451393, −0.922250256607872274745327044120, 2.01508538680269571224798095897, 3.85776435744809464175471361488, 6.57960384896072749506859310635, 7.899061322944095667950763000491, 10.32652033220571165123129081401, 11.30630258922559240201058234175, 12.41123842594293467110988486623, 14.50836010594884743943933715357, 15.48920813006426995579499060857, 16.74507245906501821586311458060

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