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Properties

Label 2-666-111.56-c1-0-6
Degree $2$
Conductor $666$
Sign $0.621 + 0.783i$
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 + (−3.66 + 1.70i)5-s + (−0.294 − 0.107i)7-s + (0.258 + 0.965i)8-s + (2.02 + 3.49i)10-s + (0.875 − 1.51i)11-s + (1.84 − 2.64i)13-s + (−0.0811 + 0.302i)14-s + (0.939 − 0.342i)16-s + (0.590 + 0.843i)17-s + (5.81 + 0.508i)19-s + (3.31 − 2.31i)20-s + (−1.58 − 0.740i)22-s + (2.42 + 0.648i)23-s + ⋯
L(s)  = 1  + (−0.0616 − 0.704i)2-s + (−0.492 + 0.0868i)4-s + (−1.63 + 0.763i)5-s + (−0.111 − 0.0405i)7-s + (0.0915 + 0.341i)8-s + (0.638 + 1.10i)10-s + (0.264 − 0.457i)11-s + (0.513 − 0.732i)13-s + (−0.0216 + 0.0809i)14-s + (0.234 − 0.0855i)16-s + (0.143 + 0.204i)17-s + (1.33 + 0.116i)19-s + (0.740 − 0.518i)20-s + (−0.338 − 0.157i)22-s + (0.504 + 0.135i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.621 + 0.783i$
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: \((2,\ 666,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905311 - 0.437220i\)
\(L(\frac12)\) \(\approx\) \(0.905311 - 0.437220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.99 + 1.00i)T \)
good5 \( 1 + (3.66 - 1.70i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (0.294 + 0.107i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.875 + 1.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.84 + 2.64i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-0.590 - 0.843i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (-5.81 - 0.508i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (-2.42 - 0.648i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.34 + 0.627i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.925 + 0.925i)T - 31iT^{2} \)
41 \( 1 + (-0.370 - 2.10i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.14 - 4.14i)T + 43iT^{2} \)
47 \( 1 + (-3.98 + 2.30i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.72 + 7.48i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.432 - 0.927i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (-0.131 - 0.0918i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (5.46 - 15.0i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (8.74 + 10.4i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 + (0.448 + 0.961i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (-8.03 - 1.41i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-12.5 - 5.83i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (-2.88 + 10.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.62308265865945400556800136307, −9.722018844519752181050071448513, −8.545082037633744685398356603490, −7.88416735109652265310226645304, −7.09550364096860882227765162010, −5.86060799961367549887608606930, −4.50559776627050540648472248816, −3.50453401558867248630444039106, −2.96959409827925382835824471310, −0.810124911606967966735056491030, 0.981275673397958697669091637760, 3.33069111067088489583429881422, 4.33500668953339006690767083666, 4.98808442216174076636393369181, 6.31415275627803459377199152254, 7.40407360043182769778600118607, 7.79808802362159615715731491967, 8.925675994676100340454760663535, 9.309750764132407728735345856756, 10.71575006884285741155690459758

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