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Properties

Label 2-17-17.16-c9-0-4
Degree $2$
Conductor $17$
Sign $-0.516 - 0.856i$
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  + 25.8·2-s + 225. i·3-s + 154.·4-s + 96.4i·5-s + 5.81e3i·6-s + 1.38e3i·7-s − 9.22e3·8-s − 3.10e4·9-s + 2.49e3i·10-s + 3.38e4i·11-s + 3.48e4i·12-s + 1.38e5·13-s + 3.57e4i·14-s − 2.17e4·15-s − 3.17e5·16-s + (1.77e5 + 2.94e5i)17-s + ⋯
L(s)  = 1  + 1.14·2-s + 1.60i·3-s + 0.302·4-s + 0.0690i·5-s + 1.83i·6-s + 0.218i·7-s − 0.796·8-s − 1.57·9-s + 0.0787i·10-s + 0.696i·11-s + 0.485i·12-s + 1.34·13-s + 0.248i·14-s − 0.110·15-s − 1.21·16-s + (0.516 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.516 - 0.856i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.25492 + 2.22352i\)
\(L(\frac12)\) \(\approx\) \(1.25492 + 2.22352i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.77e5 - 2.94e5i)T \)
good2 \( 1 - 25.8T + 512T^{2} \)
3 \( 1 - 225. iT - 1.96e4T^{2} \)
5 \( 1 - 96.4iT - 1.95e6T^{2} \)
7 \( 1 - 1.38e3iT - 4.03e7T^{2} \)
11 \( 1 - 3.38e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.38e5T + 1.06e10T^{2} \)
19 \( 1 - 4.29e5T + 3.22e11T^{2} \)
23 \( 1 + 3.52e5iT - 1.80e12T^{2} \)
29 \( 1 + 5.08e5iT - 1.45e13T^{2} \)
31 \( 1 + 8.73e6iT - 2.64e13T^{2} \)
37 \( 1 - 1.62e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.12e7iT - 3.27e14T^{2} \)
43 \( 1 + 2.43e7T + 5.02e14T^{2} \)
47 \( 1 + 1.11e7T + 1.11e15T^{2} \)
53 \( 1 - 3.54e7T + 3.29e15T^{2} \)
59 \( 1 - 5.03e7T + 8.66e15T^{2} \)
61 \( 1 - 1.53e8iT - 1.16e16T^{2} \)
67 \( 1 + 2.44e8T + 2.72e16T^{2} \)
71 \( 1 + 3.78e8iT - 4.58e16T^{2} \)
73 \( 1 - 1.38e7iT - 5.88e16T^{2} \)
79 \( 1 + 5.65e8iT - 1.19e17T^{2} \)
83 \( 1 + 5.68e8T + 1.86e17T^{2} \)
89 \( 1 - 2.61e8T + 3.50e17T^{2} \)
97 \( 1 + 1.04e9iT - 7.60e17T^{2} \)
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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

−16.67086807542993625269148968307, −15.40597795491493822170280140186, −14.78731644083838175131593335006, −13.35289992754355429388512112394, −11.71908486022706062257991033832, −10.23644690991075714738693348984, −8.845550097856706796916438367264, −5.85619878146767379171472687031, −4.51061719440444289748628591740, −3.36488073646262997266880470234, 0.980109914157138175713554947025, 3.19728578534609045569333698391, 5.56091230633203782413790025509, 6.90028123876712044213115876743, 8.618835279499712785366501784884, 11.41568620185706051552028691947, 12.56966618918404039168339444633, 13.56753107029655990512679513350, 14.20548642356358078416331606197, 16.14123043729498771055201666954

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