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Properties

Label 2-666-111.56-c1-0-5
Degree $2$
Conductor $666$
Sign $0.994 - 0.106i$
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  + (−0.0871 − 0.996i)2-s + (−0.984 + 0.173i)4-s + (0.700 − 0.326i)5-s + (4.53 + 1.65i)7-s + (0.258 + 0.965i)8-s + (−0.386 − 0.668i)10-s + (−2.42 + 4.19i)11-s + (−2.50 + 3.57i)13-s + (1.25 − 4.66i)14-s + (0.939 − 0.342i)16-s + (1.37 + 1.96i)17-s + (5.73 + 0.501i)19-s + (−0.632 + 0.443i)20-s + (4.39 + 2.04i)22-s + (−5.80 − 1.55i)23-s + ⋯
L(s)  = 1  + (−0.0616 − 0.704i)2-s + (−0.492 + 0.0868i)4-s + (0.313 − 0.145i)5-s + (1.71 + 0.624i)7-s + (0.0915 + 0.341i)8-s + (−0.122 − 0.211i)10-s + (−0.730 + 1.26i)11-s + (−0.694 + 0.992i)13-s + (0.334 − 1.24i)14-s + (0.234 − 0.0855i)16-s + (0.334 + 0.477i)17-s + (1.31 + 0.115i)19-s + (−0.141 + 0.0990i)20-s + (0.936 + 0.436i)22-s + (−1.21 − 0.324i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59032 + 0.0853091i\)
\(L(\frac12)\) \(\approx\) \(1.59032 + 0.0853091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 + 0.996i)T \)
3 \( 1 \)
37 \( 1 + (-2.35 + 5.60i)T \)
good5 \( 1 + (-0.700 + 0.326i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (-4.53 - 1.65i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (2.42 - 4.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.50 - 3.57i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-1.37 - 1.96i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (-5.73 - 0.501i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (5.80 + 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.94 + 0.520i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.21 - 2.21i)T - 31iT^{2} \)
41 \( 1 + (1.55 + 8.84i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-5.76 - 5.76i)T + 43iT^{2} \)
47 \( 1 + (-8.09 + 4.67i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.878 - 2.41i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-3.50 + 7.51i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (3.07 + 2.15i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (-4.27 + 11.7i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.82 - 8.13i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + 5.96iT - 73T^{2} \)
79 \( 1 + (5.34 + 11.4i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (7.54 + 1.32i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (8.54 + 3.98i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (0.616 - 2.30i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.54738479038107834215712968272, −9.712350186179226493230755177292, −8.999970681750513351734863541067, −7.906115259097675550756245607921, −7.37981354197648554725433295759, −5.62664756695663617447180166607, −4.97855838042070299254215779789, −4.09928220529845719730496240419, −2.28684806745712378893729677947, −1.73482657963743804581310465442, 0.951860138645242554113452201642, 2.74159571757286144782158211099, 4.20715632134125164433991710545, 5.33730080082233370093608828257, 5.71335392981183234237030337527, 7.25472978652460767740899158241, 7.935876557063147740694375924402, 8.318543722569496369108722445211, 9.718589316742069574656643484481, 10.39963147887729987211691124458

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