Properties

Label 2-666-37.34-c1-0-14
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: $\chi_{666} (145, \cdot )$
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

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

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