↑

Properties

Label 2-666-37.21-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.0112 - 0.999i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−1.76 − 0.310i)5-s + (−0.343 + 1.94i)7-s + (−0.866 + 0.500i)8-s + (−0.894 + 1.54i)10-s + (−1.99 − 3.44i)11-s + (−3.21 + 3.83i)13-s + (1.71 + 0.987i)14-s + (0.173 + 0.984i)16-s + (2.87 + 3.43i)17-s + (1.64 + 4.52i)19-s + (1.15 + 1.37i)20-s + (−3.92 + 0.691i)22-s + (−1.37 − 0.795i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.383 − 0.321i)4-s + (−0.788 − 0.138i)5-s + (−0.129 + 0.735i)7-s + (−0.306 + 0.176i)8-s + (−0.282 + 0.489i)10-s + (−0.600 − 1.03i)11-s + (−0.892 + 1.06i)13-s + (0.457 + 0.264i)14-s + (0.0434 + 0.246i)16-s + (0.698 + 0.832i)17-s + (0.377 + 1.03i)19-s + (0.257 + 0.306i)20-s + (−0.836 + 0.147i)22-s + (−0.287 − 0.165i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0112 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0112 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.0112 - 0.999i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.0112 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363207 + 0.367303i\)
\(L(\frac12)\) \(\approx\) \(0.363207 + 0.367303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 \)
37 \( 1 + (-1.03 + 5.99i)T \)
good5 \( 1 + (1.76 + 0.310i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.343 - 1.94i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.99 + 3.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.21 - 3.83i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.87 - 3.43i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-1.64 - 4.52i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (1.37 + 0.795i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.71 - 5.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.70iT - 31T^{2} \)
41 \( 1 + (0.119 + 0.100i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 5.46iT - 43T^{2} \)
47 \( 1 + (5.65 - 9.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.387 + 2.20i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-0.606 + 0.106i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.86 + 3.41i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.21 + 6.88i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.838 + 0.305i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 0.957T + 73T^{2} \)
79 \( 1 + (14.1 + 2.48i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.579 + 0.486i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (12.8 - 2.27i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (3.95 + 2.28i)T + (48.5 + 84.0i)T^{2} \)
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

−10.89777372571796556161761271093, −9.961945307803580196697255225808, −9.046800614424260934939818359768, −8.262631581211290466599835446255, −7.37978043055361355566681573085, −5.97741415078054684012014072874, −5.25318308985291785694111041106, −3.99969107809060826450793013777, −3.18180716806679185217093064297, −1.80880858801124309415309643698, 0.24656443554076467582363564326, 2.69192517218112614056792058775, 3.87807715429130107892766761706, 4.81712357169547775291431024564, 5.67225306553397572040741231537, 7.22572488609505994772253507237, 7.40262340414702402717941132060, 8.139453439339744610588183387107, 9.692905340444911001472652913124, 9.965498073983526953650674025560

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