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Properties

Label 2-666-37.12-c1-0-3
Degree $2$
Conductor $666$
Sign $0.619 - 0.785i$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−3.31 + 1.20i)5-s + (0.826 − 0.300i)7-s + (−0.500 − 0.866i)8-s + (−1.76 + 3.05i)10-s + (1.67 + 2.89i)11-s + (−1.11 + 6.31i)13-s + (0.439 − 0.761i)14-s + (−0.939 − 0.342i)16-s + (0.520 + 2.95i)17-s + (3.55 + 2.98i)19-s + (0.613 + 3.47i)20-s + (3.14 + 1.14i)22-s + (2.91 − 5.05i)23-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (0.0868 − 0.492i)4-s + (−1.48 + 0.540i)5-s + (0.312 − 0.113i)7-s + (−0.176 − 0.306i)8-s + (−0.558 + 0.967i)10-s + (0.504 + 0.874i)11-s + (−0.308 + 1.75i)13-s + (0.117 − 0.203i)14-s + (−0.234 − 0.0855i)16-s + (0.126 + 0.716i)17-s + (0.815 + 0.683i)19-s + (0.137 + 0.777i)20-s + (0.670 + 0.244i)22-s + (0.608 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.619 - 0.785i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.619 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23533 + 0.599006i\)
\(L(\frac12)\) \(\approx\) \(1.23533 + 0.599006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 \)
37 \( 1 + (3.44 - 5.01i)T \)
good5 \( 1 + (3.31 - 1.20i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.826 + 0.300i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.67 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.11 - 6.31i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.520 - 2.95i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-3.55 - 2.98i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.91 + 5.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.63 + 2.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.12T + 31T^{2} \)
41 \( 1 + (1.49 - 8.45i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + (2.56 - 4.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.252 - 0.0918i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-7.15 - 2.60i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.369 + 2.09i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (8.01 - 2.91i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (10.0 + 8.45i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.57T + 73T^{2} \)
79 \( 1 + (-10.8 + 3.96i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.724 + 4.10i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-7.86 - 2.86i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.53 + 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.89427495224354639696820923521, −9.960606342011450635141213272505, −8.985675672592111762274251977966, −7.85518562647060889728101517914, −7.09563908967999520617939661365, −6.30596252168782563256514129020, −4.61159273714614770343798943314, −4.23270763117236086800550356660, −3.16138782679095722754651403756, −1.66975853339689439329240680885, 0.65217048266542788411252693929, 3.11000203027628009556026200891, 3.76036816228542401235675350557, 5.07907031061977699291731928341, 5.51870036490835082330827731064, 7.14895271740637504805841301886, 7.62324406326175006424562192611, 8.476142706995938174948591151485, 9.202875624959745008122599609860, 10.67988653192459738567897032865

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