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Properties

Label 2-666-333.211-c1-0-19
Degree $2$
Conductor $666$
Sign $0.955 - 0.295i$
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 + (1.10 − 1.33i)3-s + (−0.499 − 0.866i)4-s + (1.14 + 1.98i)5-s + (0.600 + 1.62i)6-s + 3.77·7-s + 0.999·8-s + (−0.549 − 2.94i)9-s − 2.29·10-s + (−0.643 − 1.11i)11-s + (−1.70 − 0.292i)12-s + (0.585 + 1.01i)13-s + (−1.88 + 3.26i)14-s + (3.91 + 0.671i)15-s + (−0.5 + 0.866i)16-s + (1.33 + 2.31i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.639 − 0.769i)3-s + (−0.249 − 0.433i)4-s + (0.513 + 0.888i)5-s + (0.245 + 0.663i)6-s + 1.42·7-s + 0.353·8-s + (−0.183 − 0.983i)9-s − 0.725·10-s + (−0.194 − 0.336i)11-s + (−0.492 − 0.0844i)12-s + (0.162 + 0.281i)13-s + (−0.504 + 0.873i)14-s + (1.01 + 0.173i)15-s + (−0.125 + 0.216i)16-s + (0.323 + 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86885 + 0.282559i\)
\(L(\frac12)\) \(\approx\) \(1.86885 + 0.282559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.33i)T \)
37 \( 1 + (-2.56 + 5.51i)T \)
good5 \( 1 + (-1.14 - 1.98i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.77T + 7T^{2} \)
11 \( 1 + (0.643 + 1.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.585 - 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.33 - 2.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.52 + 2.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.30 - 5.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.54 + 2.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.05 - 1.82i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (5.21 + 9.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.40 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.25 - 9.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.47 - 9.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3.28T + 59T^{2} \)
61 \( 1 - 7.79T + 61T^{2} \)
67 \( 1 + (5.04 + 8.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.73 + 6.47i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.88T + 73T^{2} \)
79 \( 1 + 6.82T + 79T^{2} \)
83 \( 1 + (2.25 - 3.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.558 + 0.966i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.32 + 4.03i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54595894588792761187140213826, −9.448282238438012232393284853001, −8.656967863686233430264803390892, −7.76154138760869986913337366883, −7.31612928222769754829778406161, −6.21941497055202713866835906611, −5.46489702545280702472814811592, −3.96067047487151925381626745216, −2.49761719452906742268180280938, −1.43451683292578224067077702482, 1.43373905612430514573702480370, 2.49306908950129106771727203967, 3.93685478234971909258539037882, 4.87139939495682173474301306380, 5.45251546975581424410218086780, 7.38598995513261897717312208409, 8.448548282915410854323897483175, 8.547370060408526983883825365652, 9.803021174372154065351747437940, 10.19803409593042324767781293497

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