Properties

Label 2-17-17.13-c9-0-7
Degree $2$
Conductor $17$
Sign $-0.0744 + 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 35.4i·2-s + (137. + 137. i)3-s − 744.·4-s + (178. + 178. i)5-s + (4.88e3 − 4.88e3i)6-s + (7.49e3 − 7.49e3i)7-s + 8.25e3i·8-s + 1.83e4i·9-s + (6.31e3 − 6.31e3i)10-s + (2.77e4 − 2.77e4i)11-s + (−1.02e5 − 1.02e5i)12-s − 2.08e4·13-s + (−2.65e5 − 2.65e5i)14-s + 4.90e4i·15-s − 8.87e4·16-s + (2.86e5 − 1.91e5i)17-s + ⋯
L(s)  = 1  − 1.56i·2-s + (0.982 + 0.982i)3-s − 1.45·4-s + (0.127 + 0.127i)5-s + (1.53 − 1.53i)6-s + (1.17 − 1.17i)7-s + 0.712i·8-s + 0.930i·9-s + (0.199 − 0.199i)10-s + (0.571 − 0.571i)11-s + (−1.42 − 1.42i)12-s − 0.202·13-s + (−1.84 − 1.84i)14-s + 0.250i·15-s − 0.338·16-s + (0.831 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0744 + 0.997i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0744 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.0744 + 0.997i$
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: \((2,\ 17,\ (\ :9/2),\ -0.0744 + 0.997i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.66878 - 1.79805i\)
\(L(\frac12)\) \(\approx\) \(1.66878 - 1.79805i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.86e5 + 1.91e5i)T \)
good2 \( 1 + 35.4iT - 512T^{2} \)
3 \( 1 + (-137. - 137. i)T + 1.96e4iT^{2} \)
5 \( 1 + (-178. - 178. i)T + 1.95e6iT^{2} \)
7 \( 1 + (-7.49e3 + 7.49e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (-2.77e4 + 2.77e4i)T - 2.35e9iT^{2} \)
13 \( 1 + 2.08e4T + 1.06e10T^{2} \)
19 \( 1 - 8.88e5iT - 3.22e11T^{2} \)
23 \( 1 + (8.39e5 - 8.39e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (-1.43e6 - 1.43e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-7.10e5 - 7.10e5i)T + 2.64e13iT^{2} \)
37 \( 1 + (8.16e6 + 8.16e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (1.89e7 - 1.89e7i)T - 3.27e14iT^{2} \)
43 \( 1 - 3.58e7iT - 5.02e14T^{2} \)
47 \( 1 + 1.67e7T + 1.11e15T^{2} \)
53 \( 1 - 8.53e7iT - 3.29e15T^{2} \)
59 \( 1 + 4.95e5iT - 8.66e15T^{2} \)
61 \( 1 + (-5.17e7 + 5.17e7i)T - 1.16e16iT^{2} \)
67 \( 1 + 2.90e8T + 2.72e16T^{2} \)
71 \( 1 + (-2.15e8 - 2.15e8i)T + 4.58e16iT^{2} \)
73 \( 1 + (7.17e7 + 7.17e7i)T + 5.88e16iT^{2} \)
79 \( 1 + (-2.24e8 + 2.24e8i)T - 1.19e17iT^{2} \)
83 \( 1 - 4.86e7iT - 1.86e17T^{2} \)
89 \( 1 - 6.13e7T + 3.50e17T^{2} \)
97 \( 1 + (8.46e8 + 8.46e8i)T + 7.60e17iT^{2} \)
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

−16.42526075300025454774037557064, −14.39743830282024460825607727888, −13.93207482281679012747417863996, −11.89497818989520937103386297614, −10.56936076882839148995766822045, −9.747538544374669371741915285550, −8.153215719327348015120511260682, −4.38453002381063923010743848530, −3.32161141310045634901807743769, −1.38734610221376535367551368172, 2.00356607764643996321452199950, 5.17146728922112577990438282544, 6.90379825150481689432251204611, 8.142988573426123801689170007756, 8.937475245935342285568898864607, 12.04091408009476882438144173338, 13.61069210041930094177609927642, 14.67127187999309946757234253653, 15.30949211294132294702147287751, 17.20009449487259556188136851134

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