↑

Properties

Label 2-17-17.16-c9-0-5
Degree $2$
Conductor $17$
Sign $0.894 + 0.448i$
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  − 9.15·2-s + 105. i·3-s − 428.·4-s − 1.75e3i·5-s − 968. i·6-s + 5.86e3i·7-s + 8.60e3·8-s + 8.49e3·9-s + 1.60e4i·10-s − 5.72e4i·11-s − 4.52e4i·12-s + 1.83e5·13-s − 5.37e4i·14-s + 1.85e5·15-s + 1.40e5·16-s + (−3.07e5 − 1.54e5i)17-s + ⋯
L(s)  = 1  − 0.404·2-s + 0.753i·3-s − 0.836·4-s − 1.25i·5-s − 0.304i·6-s + 0.924i·7-s + 0.742·8-s + 0.431·9-s + 0.507i·10-s − 1.17i·11-s − 0.630i·12-s + 1.77·13-s − 0.373i·14-s + 0.945·15-s + 0.535·16-s + (−0.894 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.448i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.894 + 0.448i$
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: \((2,\ 17,\ (\ :9/2),\ 0.894 + 0.448i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.13725 - 0.269026i\)
\(L(\frac12)\) \(\approx\) \(1.13725 - 0.269026i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.07e5 + 1.54e5i)T \)
good2 \( 1 + 9.15T + 512T^{2} \)
3 \( 1 - 105. iT - 1.96e4T^{2} \)
5 \( 1 + 1.75e3iT - 1.95e6T^{2} \)
7 \( 1 - 5.86e3iT - 4.03e7T^{2} \)
11 \( 1 + 5.72e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.83e5T + 1.06e10T^{2} \)
19 \( 1 - 2.69e5T + 3.22e11T^{2} \)
23 \( 1 + 1.82e6iT - 1.80e12T^{2} \)
29 \( 1 + 1.33e6iT - 1.45e13T^{2} \)
31 \( 1 - 5.82e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.79e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.95e7iT - 3.27e14T^{2} \)
43 \( 1 - 1.23e7T + 5.02e14T^{2} \)
47 \( 1 - 3.64e7T + 1.11e15T^{2} \)
53 \( 1 + 4.39e7T + 3.29e15T^{2} \)
59 \( 1 - 6.25e7T + 8.66e15T^{2} \)
61 \( 1 - 1.82e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.37e8T + 2.72e16T^{2} \)
71 \( 1 - 5.84e7iT - 4.58e16T^{2} \)
73 \( 1 + 1.40e8iT - 5.88e16T^{2} \)
79 \( 1 + 1.26e8iT - 1.19e17T^{2} \)
83 \( 1 - 9.17e7T + 1.86e17T^{2} \)
89 \( 1 + 5.55e8T + 3.50e17T^{2} \)
97 \( 1 + 6.97e8iT - 7.60e17T^{2} \)
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

−16.35855177145799798932475921475, −15.84904527262592743602175406898, −13.81075541166723876262127970651, −12.66959960837938908847981828033, −10.77917989376835224611284276353, −8.924046913279254109433277900104, −8.743897385662560489488728996679, −5.49535396511526509017902092720, −4.09640182754312549871111592979, −0.846309112302970345245977670764, 1.34885839633621012970199457379, 3.97221310213751672417740183761, 6.69175767105858679107889977219, 7.81284594028677730885701790446, 9.808115917721173641712512479219, 11.01430902984881315234953542036, 13.12609091337998486197775275995, 13.84340458556390903534166882833, 15.42340964844686210014994034454, 17.40599294384045414570757205193

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