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Properties

Label 2-666-37.34-c1-0-8
Degree $2$
Conductor $666$
Sign $0.420 + 0.907i$
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 + (1.43 + 0.524i)5-s + (−1.82 − 0.665i)7-s + (0.500 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (0.220 − 0.381i)11-s + (−0.367 − 2.08i)13-s + (0.972 + 1.68i)14-s + (−0.939 + 0.342i)16-s + (0.664 − 3.76i)17-s + (4.55 − 3.82i)19-s + (−0.266 + 1.50i)20-s + (−0.414 + 0.150i)22-s + (2.85 + 4.94i)23-s + ⋯
L(s)  = 1  + (−0.541 − 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.643 + 0.234i)5-s + (−0.691 − 0.251i)7-s + (0.176 − 0.306i)8-s + (−0.242 − 0.419i)10-s + (0.0664 − 0.115i)11-s + (−0.101 − 0.578i)13-s + (0.260 + 0.450i)14-s + (−0.234 + 0.0855i)16-s + (0.161 − 0.913i)17-s + (1.04 − 0.876i)19-s + (−0.0594 + 0.337i)20-s + (−0.0883 + 0.0321i)22-s + (0.595 + 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.420 + 0.907i$
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: \((2,\ 666,\ (\ :1/2),\ 0.420 + 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980636 - 0.626320i\)
\(L(\frac12)\) \(\approx\) \(0.980636 - 0.626320i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 5.67i)T \)
good5 \( 1 + (-1.43 - 0.524i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.82 + 0.665i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.220 + 0.381i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.367 + 2.08i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.664 + 3.76i)T + (-15.9 - 5.81i)T^{2} \)
19 \( 1 + (-4.55 + 3.82i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-2.85 - 4.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.87T + 31T^{2} \)
41 \( 1 + (1.80 + 10.2i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 - 3.84T + 43T^{2} \)
47 \( 1 + (-3.84 - 6.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.62 + 0.591i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.93 + 0.702i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.89 + 10.7i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.57 - 0.572i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.70 + 6.46i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + 9.88T + 73T^{2} \)
79 \( 1 + (12.5 + 4.56i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (0.617 - 3.50i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (11.1 - 4.05i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-0.296 - 0.513i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.01078700136249726088911962230, −9.795987108826270587832156785260, −8.879174482132960421207092686049, −7.73101673363376485891729652687, −6.95661446753255084842516542377, −5.96422824201308203017324427736, −4.83546241861287634090547844414, −3.33758183776374253698779632411, −2.57161638740323803720450183651, −0.846528745226404178410237982733, 1.35849454081504879462978043221, 2.79273345163135643029196320003, 4.28995126154914898583971191004, 5.55709606558427356604774142827, 6.22803430876508172699941065040, 7.09251472376916860090824807021, 8.188593767390181663298517606290, 8.975800943646751046785440758382, 9.816782761800446976204752320714, 10.24559206493093175974204669402

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