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Properties

Label 2-666-333.232-c1-0-35
Degree $2$
Conductor $666$
Sign $0.702 + 0.711i$
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  + (0.5 + 0.866i)2-s + (0.211 − 1.71i)3-s + (−0.499 + 0.866i)4-s + (2.03 − 3.53i)5-s + (1.59 − 0.676i)6-s + 4.94·7-s − 0.999·8-s + (−2.91 − 0.728i)9-s + 4.07·10-s + (−1.34 + 2.33i)11-s + (1.38 + 1.04i)12-s + (1.43 − 2.48i)13-s + (2.47 + 4.28i)14-s + (−5.63 − 4.25i)15-s + (−0.5 − 0.866i)16-s + (−2.71 + 4.71i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.122 − 0.992i)3-s + (−0.249 + 0.433i)4-s + (0.912 − 1.57i)5-s + (0.651 − 0.275i)6-s + 1.87·7-s − 0.353·8-s + (−0.970 − 0.242i)9-s + 1.28·10-s + (−0.406 + 0.703i)11-s + (0.399 + 0.301i)12-s + (0.398 − 0.690i)13-s + (0.661 + 1.14i)14-s + (−1.45 − 1.09i)15-s + (−0.125 − 0.216i)16-s + (−0.659 + 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17676 - 0.909148i\)
\(L(\frac12)\) \(\approx\) \(2.17676 - 0.909148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.211 + 1.71i)T \)
37 \( 1 + (6.08 - 0.0307i)T \)
good5 \( 1 + (-2.03 + 3.53i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.94T + 7T^{2} \)
11 \( 1 + (1.34 - 2.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.43 + 2.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.71 - 4.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.179 - 0.311i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.147 - 0.256i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.72 - 2.99i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.980 - 1.69i)T + (-15.5 + 26.8i)T^{2} \)
41 \( 1 + (1.23 - 2.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.92 + 6.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.83 - 6.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.45 + 4.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.55T + 59T^{2} \)
61 \( 1 - 4.14T + 61T^{2} \)
67 \( 1 + (-0.283 + 0.490i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.34 - 9.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.59T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 + (6.24 + 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.19 - 15.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.696 + 1.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44417584168194956054759669088, −9.060937638147034801635084439773, −8.320555989874720951766727223201, −8.082336720115466202523971066609, −6.88798911969858048620007146017, −5.57941507459694958517019367331, −5.27771176157317925420382337769, −4.25193285091975466631509519073, −2.10668632809718183658108918693, −1.31428343357699851740514620729, 2.02553767975145178565742745969, 2.82907393545761887225578577000, 4.03490927265326734254604034681, 5.07648522083286646601112688138, 5.80109068349176291462237671545, 6.99468196004762183682693292973, 8.226001364915122823251115362247, 9.148165722758743972618602118780, 10.03591607108970701723273487594, 10.82484404195229326589175369312

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