Properties

Label 2-666-37.4-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.771 - 0.636i$
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  + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (0.247 + 0.294i)5-s + (−2.50 + 2.09i)7-s + (−0.866 − 0.5i)8-s + (−0.192 − 0.333i)10-s + (1.29 − 2.23i)11-s + (−0.466 + 1.28i)13-s + (2.82 − 1.63i)14-s + (0.766 + 0.642i)16-s + (1.13 + 3.13i)17-s + (−5.89 + 1.03i)19-s + (0.131 + 0.361i)20-s + (−1.66 + 1.98i)22-s + (−6.53 + 3.77i)23-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.469 + 0.171i)4-s + (0.110 + 0.131i)5-s + (−0.945 + 0.793i)7-s + (−0.306 − 0.176i)8-s + (−0.0608 − 0.105i)10-s + (0.389 − 0.674i)11-s + (−0.129 + 0.355i)13-s + (0.755 − 0.436i)14-s + (0.191 + 0.160i)16-s + (0.276 + 0.759i)17-s + (−1.35 + 0.238i)19-s + (0.0294 + 0.0808i)20-s + (−0.354 + 0.422i)22-s + (−1.36 + 0.786i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136688 + 0.380479i\)
\(L(\frac12)\) \(\approx\) \(0.136688 + 0.380479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 \)
37 \( 1 + (-0.543 - 6.05i)T \)
good5 \( 1 + (-0.247 - 0.294i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.50 - 2.09i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.29 + 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.466 - 1.28i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.13 - 3.13i)T + (-13.0 + 10.9i)T^{2} \)
19 \( 1 + (5.89 - 1.03i)T + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (6.53 - 3.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.78 + 1.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
41 \( 1 + (7.77 + 2.82i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 4.33iT - 43T^{2} \)
47 \( 1 + (-2.61 - 4.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.64 - 5.57i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (8.33 - 9.93i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.39 - 6.58i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.60 - 7.22i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.60 + 14.7i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + (-0.940 - 1.12i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.0104 - 0.00380i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-0.612 + 0.730i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-12.8 + 7.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.51001923864513220039304290337, −10.09871479281111193400956306863, −8.973177352374044066003950988477, −8.579177173110542916355767760859, −7.44481206551226673142329743857, −6.15153483007323053819535857062, −6.03693246569663202223498527866, −4.15418958640919460799055541564, −3.04664155857620972791245791439, −1.84779077836505118556444181594, 0.25389976728869245259753105109, 2.00626002608542527129342900166, 3.43684369406898987111630978775, 4.57338916692161725576830802087, 5.93923304243480058848893629135, 6.81194332456490766913817735883, 7.46419126543612011807708835003, 8.523014295225440856566956149062, 9.466393202715480604099469933275, 10.03833362879039776503982747594

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