↑

Properties

Label 2-17-17.16-c7-0-4
Degree $2$
Conductor $17$
Sign $0.636 + 0.771i$
Analytic cond. $5.31054$
Root an. cond. $2.30446$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 11.2·2-s + 9.73i·3-s − 0.992·4-s + 181. i·5-s − 109. i·6-s − 1.34e3i·7-s + 1.45e3·8-s + 2.09e3·9-s − 2.04e3i·10-s − 5.78e3i·11-s − 9.66i·12-s + 834.·13-s + 1.51e4i·14-s − 1.76e3·15-s − 1.62e4·16-s + (1.28e4 + 1.56e4i)17-s + ⋯
L(s)  = 1  − 0.996·2-s + 0.208i·3-s − 0.00775·4-s + 0.649i·5-s − 0.207i·6-s − 1.47i·7-s + 1.00·8-s + 0.956·9-s − 0.646i·10-s − 1.31i·11-s − 0.00161i·12-s + 0.105·13-s + 1.47i·14-s − 0.135·15-s − 0.992·16-s + (0.636 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.636 + 0.771i$
Analytic conductor: \(5.31054\)
Root analytic conductor: \(2.30446\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :7/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.751934 - 0.354567i\)
\(L(\frac12)\) \(\approx\) \(0.751934 - 0.354567i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.28e4 - 1.56e4i)T \)
good2 \( 1 + 11.2T + 128T^{2} \)
3 \( 1 - 9.73iT - 2.18e3T^{2} \)
5 \( 1 - 181. iT - 7.81e4T^{2} \)
7 \( 1 + 1.34e3iT - 8.23e5T^{2} \)
11 \( 1 + 5.78e3iT - 1.94e7T^{2} \)
13 \( 1 - 834.T + 6.27e7T^{2} \)
19 \( 1 + 9.26e3T + 8.93e8T^{2} \)
23 \( 1 + 1.07e5iT - 3.40e9T^{2} \)
29 \( 1 + 1.01e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.09e4iT - 2.75e10T^{2} \)
37 \( 1 - 3.01e4iT - 9.49e10T^{2} \)
41 \( 1 + 3.95e5iT - 1.94e11T^{2} \)
43 \( 1 + 5.64e5T + 2.71e11T^{2} \)
47 \( 1 - 5.02e5T + 5.06e11T^{2} \)
53 \( 1 - 2.53e5T + 1.17e12T^{2} \)
59 \( 1 - 2.58e6T + 2.48e12T^{2} \)
61 \( 1 + 1.00e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.93e6T + 6.06e12T^{2} \)
71 \( 1 - 3.95e6iT - 9.09e12T^{2} \)
73 \( 1 - 5.07e6iT - 1.10e13T^{2} \)
79 \( 1 + 8.11e5iT - 1.92e13T^{2} \)
83 \( 1 + 3.58e6T + 2.71e13T^{2} \)
89 \( 1 - 5.31e6T + 4.42e13T^{2} \)
97 \( 1 - 1.41e7iT - 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.07759327849233528119213968768, −16.30515715396954300368671614840, −14.38057708371840032062283547989, −13.21990863161805717669246145157, −10.71954911071896861341715428985, −10.22602398691951265597756767005, −8.348261571822217970843324421286, −6.92193708839619448350575842274, −4.04268609371314312419527375998, −0.808962553715766202230412162123, 1.58395072283930720783249062906, 4.97407593773395745927101603282, 7.44074743490386239532706975171, 8.966230288667469646439331398768, 9.872835362564853783678421787622, 12.02425720299827284560838932337, 13.13485370750890817549013119107, 15.16112607139964541503384824698, 16.34825177849778489691775220282, 17.83529615057325608643477176859

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