↑

Properties

Label 2-666-37.33-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.993 - 0.116i$
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  + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.266 + 0.223i)5-s + (0.773 − 0.649i)7-s + (0.5 − 0.866i)8-s + (−0.173 − 0.300i)10-s + (−2.73 + 4.73i)11-s + (−5.97 − 2.17i)13-s + (0.504 + 0.874i)14-s + (0.766 + 0.642i)16-s + (0.490 − 0.178i)17-s + (0.869 + 4.93i)19-s + (0.326 − 0.118i)20-s + (−4.18 − 3.51i)22-s + (0.721 + 1.24i)23-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.469 − 0.171i)4-s + (−0.118 + 0.0998i)5-s + (0.292 − 0.245i)7-s + (0.176 − 0.306i)8-s + (−0.0549 − 0.0951i)10-s + (−0.823 + 1.42i)11-s + (−1.65 − 0.603i)13-s + (0.134 + 0.233i)14-s + (0.191 + 0.160i)16-s + (0.119 − 0.0433i)17-s + (0.199 + 1.13i)19-s + (0.0729 − 0.0265i)20-s + (−0.892 − 0.748i)22-s + (0.150 + 0.260i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0363591 + 0.622208i\)
\(L(\frac12)\) \(\approx\) \(0.0363591 + 0.622208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 \)
37 \( 1 + (5.88 + 1.53i)T \)
good5 \( 1 + (0.266 - 0.223i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.773 + 0.649i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.97 + 2.17i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.490 + 0.178i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.869 - 4.93i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-0.721 - 1.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.16 - 5.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
41 \( 1 + (1.03 + 0.376i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + (5.71 + 9.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0706 + 0.0593i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.29 + 1.92i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.58 + 1.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.892 - 0.748i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.0543 + 0.308i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (-7.12 + 5.98i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.89 + 3.23i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-12.0 - 10.1i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.464 - 0.804i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.58154902973708656803735098506, −10.02523052701226341003352747413, −9.287759703574053876117740828011, −7.904278651939843318347126154078, −7.56840114755383099172095847984, −6.76143807509163166107939383912, −5.23111670896196396336389331537, −4.95540311898102083933506099915, −3.47605580496576728694807373637, −1.93423725160281862749324298382, 0.32879356204538299275772602722, 2.24094727434443105193779167700, 3.12588525505060107679535247726, 4.56637308583442742776657776057, 5.26180378173184383738691248700, 6.53229336041695525501882289261, 7.74765096172053458332467390439, 8.422914127235538523452172949660, 9.354983001861693518880338574403, 10.13556759166392988609182837037

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