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Properties

Label 2-666-37.30-c1-0-0
Degree $2$
Conductor $666$
Sign $-0.00839 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.839 + 0.148i)5-s + (0.240 + 1.36i)7-s + (0.866 + 0.500i)8-s + (0.426 + 0.738i)10-s + (−0.466 + 0.807i)11-s + (−2.34 − 2.78i)13-s + (1.19 − 0.692i)14-s + (0.173 − 0.984i)16-s + (−2.84 + 3.39i)17-s + (−1.30 + 3.59i)19-s + (0.548 − 0.653i)20-s + (0.917 + 0.161i)22-s + (−0.920 + 0.531i)23-s + ⋯
L(s)  = 1  + (−0.241 − 0.664i)2-s + (−0.383 + 0.321i)4-s + (−0.375 + 0.0662i)5-s + (0.0908 + 0.515i)7-s + (0.306 + 0.176i)8-s + (0.134 + 0.233i)10-s + (−0.140 + 0.243i)11-s + (−0.649 − 0.773i)13-s + (0.320 − 0.185i)14-s + (0.0434 − 0.246i)16-s + (−0.690 + 0.822i)17-s + (−0.300 + 0.824i)19-s + (0.122 − 0.146i)20-s + (0.195 + 0.0345i)22-s + (−0.191 + 0.110i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00839 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00839 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.00839 - 0.999i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.00839 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396345 + 0.399686i\)
\(L(\frac12)\) \(\approx\) \(0.396345 + 0.399686i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 5.97i)T \)
good5 \( 1 + (0.839 - 0.148i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.240 - 1.36i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.466 - 0.807i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.34 + 2.78i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.84 - 3.39i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (1.30 - 3.59i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.920 - 0.531i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.873 + 0.504i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.33iT - 31T^{2} \)
41 \( 1 + (-0.186 + 0.156i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 5.13iT - 43T^{2} \)
47 \( 1 + (3.89 + 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.25 - 12.7i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-9.61 - 1.69i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-0.255 - 0.304i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (2.47 + 14.0i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (12.8 + 4.67i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + (-3.43 + 0.605i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-12.8 - 10.7i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (6.19 + 1.09i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (-6.47 + 3.73i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.59646362637318398068381388658, −10.08539819271543501105011190468, −9.031954724247324612386470197008, −8.221582968358414848374949252506, −7.50727162809690805472596628810, −6.22368742082123183584183628194, −5.14479437744885005774572961620, −4.06978494532524037998085513246, −2.96423298236873727139509668058, −1.75832955285474708364841631685, 0.31584257668796551679184125376, 2.31780008317526002909543174989, 4.01357360106456470820548551967, 4.73516600930952929796458566362, 5.90999261687625465518430783529, 6.99453009864851457425240714783, 7.49570963619166040843218535266, 8.532638468821681716222740794798, 9.328585579009970724296805124674, 10.11989933092702187815864788062

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