↑

Properties

Label 2-416-52.3-c0-0-1
Degree $2$
Conductor $416$
Sign $0.869 + 0.494i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 0.999·21-s + (−0.866 + 0.5i)23-s − 25-s + i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + 13-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s − 0.999·21-s + (−0.866 + 0.5i)23-s − 25-s + i·27-s + (0.5 + 0.866i)29-s + (0.499 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 0.869 + 0.494i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016835040\)
\(L(\frac12)\) \(\approx\) \(1.016835040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−11.22953263066515072973263516924, −10.52120234297227686209109891925, −9.278732705497393728421628713913, −8.658177522756836334860803512222, −7.77131598109035245563120524235, −6.66415214941330116165408095173, −5.96317286444364306479759053007, −4.09853388883791422399930173828, −3.29670150837080691439743886181, −1.79150581374722169523346327457, 2.28266084185552703061708015507, 3.53034927906849868206101622665, 4.31431976680478807299938074315, 6.01198635240913829305323751910, 6.66474908410843033072042329246, 8.157227439348933907829570435057, 8.870630352424571040050027159609, 9.597400423543448633942747884557, 10.28837658274687258026633554452, 11.67246979525751465480309726986

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