↑

Properties

Label 2-418-11.3-c1-0-5
Degree $2$
Conductor $418$
Sign $0.782 - 0.622i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (3.11 + 2.26i)5-s + (0.190 − 0.587i)7-s + (0.309 + 0.951i)8-s + (2.42 − 1.76i)9-s − 3.85·10-s + (2.54 − 2.12i)11-s + (−2 + 1.45i)13-s + (0.190 + 0.587i)14-s + (−0.809 − 0.587i)16-s + (−3.30 − 2.40i)17-s + (−0.927 + 2.85i)18-s + (0.309 + 0.951i)19-s + (3.11 − 2.26i)20-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.154 − 0.475i)4-s + (1.39 + 1.01i)5-s + (0.0721 − 0.222i)7-s + (0.109 + 0.336i)8-s + (0.809 − 0.587i)9-s − 1.21·10-s + (0.767 − 0.641i)11-s + (−0.554 + 0.403i)13-s + (0.0510 + 0.157i)14-s + (−0.202 − 0.146i)16-s + (−0.802 − 0.583i)17-s + (−0.218 + 0.672i)18-s + (0.0708 + 0.218i)19-s + (0.697 − 0.506i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32384 + 0.462559i\)
\(L(\frac12)\) \(\approx\) \(1.32384 + 0.462559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-2.54 + 2.12i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-3.11 - 2.26i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.190 + 0.587i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2 - 1.45i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.30 + 2.40i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 - 1.14T + 23T^{2} \)
29 \( 1 + (-0.381 + 1.17i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.61 + 3.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.85 - 8.78i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.23 - 9.95i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.61T + 43T^{2} \)
47 \( 1 + (3.59 + 11.0i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.61 - 1.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.09 - 9.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.16 + 5.93i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + (6.85 + 4.97i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.61 - 8.05i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.70 + 7.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.11 + 2.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-6.09 + 4.42i)T + (29.9 - 92.2i)T^{2} \)
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

−11.07633980504441421448943923918, −10.06505624439230841087416312497, −9.655394544446680966702731259253, −8.782721384707271700484804160214, −7.32681208293523091011372823298, −6.59220376276754030061246984192, −6.06223837085238675475656327415, −4.57811945031879680456433681279, −2.91907774396294469003221831187, −1.52038901077431140202826446830, 1.44079572516520030742125793663, 2.31451592277990591841759347255, 4.32111687798624885735925734803, 5.21510612809109219303831221242, 6.44109758146621749695206065768, 7.51145513266564451104704919384, 8.769581680394432977253672972722, 9.288783424634878777037911378874, 10.11026825292495302717116753788, 10.79426311943852071813024049604

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