↑

Properties

Label 2-1917-1917.283-c0-0-0
Degree $2$
Conductor $1917$
Sign $-0.830 - 0.556i$
Analytic cond. $0.956707$
Root an. cond. $0.978114$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.305 + 1.72i)2-s + (−0.411 − 0.911i)3-s + (−1.95 + 0.713i)4-s + (−1.12 − 0.942i)5-s + (1.45 − 0.989i)6-s + (−0.952 − 1.65i)8-s + (−0.661 + 0.749i)9-s + (1.28 − 2.23i)10-s + (1.45 + 1.49i)12-s + (−0.397 + 1.41i)15-s + (0.967 − 0.811i)16-s + (−1.49 − 0.915i)18-s + (0.853 + 1.47i)19-s + (2.87 + 1.04i)20-s + (−1.11 + 1.54i)24-s + (0.199 + 1.13i)25-s + ⋯
L(s)  = 1  + (0.305 + 1.72i)2-s + (−0.411 − 0.911i)3-s + (−1.95 + 0.713i)4-s + (−1.12 − 0.942i)5-s + (1.45 − 0.989i)6-s + (−0.952 − 1.65i)8-s + (−0.661 + 0.749i)9-s + (1.28 − 2.23i)10-s + (1.45 + 1.49i)12-s + (−0.397 + 1.41i)15-s + (0.967 − 0.811i)16-s + (−1.49 − 0.915i)18-s + (0.853 + 1.47i)19-s + (2.87 + 1.04i)20-s + (−1.11 + 1.54i)24-s + (0.199 + 1.13i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1917 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1917 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1917\)    =    \(3^{3} \cdot 71\)
Sign: $-0.830 - 0.556i$
Analytic conductor: \(0.956707\)
Root analytic conductor: \(0.978114\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1917} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1917,\ (\ :0),\ -0.830 - 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6195987197\)
\(L(\frac12)\) \(\approx\) \(0.6195987197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.411 + 0.911i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.305 - 1.72i)T + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (1.12 + 0.942i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.853 - 1.47i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.345 - 1.95i)T + (-0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.190 - 0.159i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.331 - 1.88i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.343 + 1.94i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.698 + 1.20i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 + 0.984i)T^{2} \)
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

−9.114569507606036224327234163032, −8.449841314993708591391405266108, −7.954793153124985634631707963403, −7.34928674481066046648814015463, −6.70603683987467376216632313834, −5.71935807573373129109252156818, −5.18811061471849420706788939057, −4.38455548596680230660190284755, −3.37343069616517965171315955419, −1.25840378046327832995184000187, 0.49861061401061639009652282549, 2.48138690473289842701821902911, 3.19666713945725853353240477514, 3.91957288158738804834802570720, 4.54694159275046161516239978999, 5.42480852636685973288296021601, 6.61739682199810242318241984181, 7.61332660219320126398777842858, 8.706186745269537373495027811139, 9.465698697799945442584926389797

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