Properties

Label 2-666-37.25-c1-0-5
Degree $2$
Conductor $666$
Sign $0.156 - 0.987i$
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  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.273 − 0.751i)5-s + (−0.138 + 0.0503i)7-s + (−0.866 + 0.500i)8-s + (0.400 − 0.692i)10-s + (2.40 + 4.16i)11-s + (1.91 + 0.338i)13-s + (−0.127 − 0.0736i)14-s + (−0.939 − 0.342i)16-s + (3.43 − 0.606i)17-s + (−4.33 + 5.16i)19-s + (0.787 − 0.138i)20-s + (−1.64 + 4.52i)22-s + (3.61 + 2.08i)23-s + ⋯
L(s)  = 1  + (0.454 + 0.541i)2-s + (−0.0868 + 0.492i)4-s + (−0.122 − 0.336i)5-s + (−0.0523 + 0.0190i)7-s + (−0.306 + 0.176i)8-s + (0.126 − 0.219i)10-s + (0.725 + 1.25i)11-s + (0.532 + 0.0938i)13-s + (−0.0341 − 0.0196i)14-s + (−0.234 − 0.0855i)16-s + (0.833 − 0.147i)17-s + (−0.994 + 1.18i)19-s + (0.176 − 0.0310i)20-s + (−0.350 + 0.963i)22-s + (0.754 + 0.435i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.156 - 0.987i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41806 + 1.21133i\)
\(L(\frac12)\) \(\approx\) \(1.41806 + 1.21133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 \)
37 \( 1 + (-6.08 - 0.0772i)T \)
good5 \( 1 + (0.273 + 0.751i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (0.138 - 0.0503i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.40 - 4.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.91 - 0.338i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.43 + 0.606i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (4.33 - 5.16i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.61 - 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.63 - 1.51i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.13iT - 31T^{2} \)
41 \( 1 + (0.676 - 3.83i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 + (-4.16 + 7.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.2 + 3.72i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-2.96 + 8.14i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.346 + 0.0610i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (2.25 - 0.820i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.34 + 4.48i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 1.13T + 73T^{2} \)
79 \( 1 + (-0.646 - 1.77i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.71 - 9.75i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-3.45 + 9.50i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.08 - 2.93i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.69424491474707361092186698483, −9.746973798474612780837181264887, −8.860294958957919633607058466971, −8.033747071509568735430665393831, −7.06208401373089977437162133911, −6.31624063576288031070517688953, −5.20809415048885213643680328680, −4.32983398504700614223963137666, −3.35748695309881018635750581864, −1.63815673782269844590108289842, 0.972805713065164850700429362278, 2.69443190832485192777184981262, 3.58878892668203832392973750617, 4.60363084241200655633591821297, 5.89066995810840123930972985512, 6.47577424171758912899962251568, 7.71218704161449701115407483203, 8.803794844810292835967477949283, 9.442740526148020917105166356869, 10.71071589751643167334545474596

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