↑

Properties

Label 2-666-333.322-c1-0-28
Degree $2$
Conductor $666$
Sign $0.174 + 0.984i$
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  + 2-s + (−1.09 + 1.34i)3-s + 4-s − 2.21·5-s + (−1.09 + 1.34i)6-s + (−0.959 − 1.66i)7-s + 8-s + (−0.606 − 2.93i)9-s − 2.21·10-s + (2.04 − 3.54i)11-s + (−1.09 + 1.34i)12-s − 3.06·13-s + (−0.959 − 1.66i)14-s + (2.42 − 2.98i)15-s + 16-s + (0.168 + 0.292i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.631 + 0.775i)3-s + 0.5·4-s − 0.992·5-s + (−0.446 + 0.548i)6-s + (−0.362 − 0.627i)7-s + 0.353·8-s + (−0.202 − 0.979i)9-s − 0.702·10-s + (0.617 − 1.06i)11-s + (−0.315 + 0.387i)12-s − 0.850·13-s + (−0.256 − 0.443i)14-s + (0.627 − 0.769i)15-s + 0.250·16-s + (0.0408 + 0.0708i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.174 + 0.984i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787694 - 0.660642i\)
\(L(\frac12)\) \(\approx\) \(0.787694 - 0.660642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.09 - 1.34i)T \)
37 \( 1 + (5.92 + 1.38i)T \)
good5 \( 1 + 2.21T + 5T^{2} \)
7 \( 1 + (0.959 + 1.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.04 + 3.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.06T + 13T^{2} \)
17 \( 1 + (-0.168 - 0.292i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 + 2.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.40 + 2.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.39 + 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 - 8.99T + 41T^{2} \)
43 \( 1 + (4.72 - 8.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.33 + 4.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.87 + 10.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.01 - 6.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.36 - 4.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 7.36T + 67T^{2} \)
71 \( 1 + (3.24 - 5.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.25T + 73T^{2} \)
79 \( 1 + (1.78 + 3.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + (-3.28 - 5.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.200 - 0.348i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.47317863162879575854788311776, −9.724441534710348061170773717320, −8.585627603057782634683204905591, −7.50467051767223085942796665021, −6.57882625181776200189937581592, −5.72997674221040620133106720709, −4.53017688855683803632009386467, −3.96290210867253512300015307978, −3.02157748520380606435554856383, −0.46931912085647812213154256608, 1.72352781844459851880559098412, 3.08090229301313781558194809563, 4.37508932997261861828515182810, 5.23864754998730345356771825518, 6.24670242381518199369997681219, 7.21209719899825087067486354481, 7.61760287011725099138421127097, 8.881707809419062688644405020951, 10.04602566821187467502558777572, 11.02028635147613976800338682325

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