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Properties

Label 2-666-111.2-c1-0-0
Degree $2$
Conductor $666$
Sign $-0.984 + 0.175i$
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 + (−1.35 − 0.630i)5-s + (−0.533 + 0.194i)7-s + (0.258 − 0.965i)8-s + (0.745 − 1.29i)10-s + (2.06 + 3.58i)11-s + (−1.18 − 1.69i)13-s + (−0.146 − 0.548i)14-s + (0.939 + 0.342i)16-s + (−3.83 + 5.46i)17-s + (0.576 − 0.0503i)19-s + (1.22 + 0.855i)20-s + (−3.75 + 1.74i)22-s + (−8.94 + 2.39i)23-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (−0.492 − 0.0868i)4-s + (−0.604 − 0.281i)5-s + (−0.201 + 0.0733i)7-s + (0.0915 − 0.341i)8-s + (0.235 − 0.408i)10-s + (0.623 + 1.08i)11-s + (−0.329 − 0.471i)13-s + (−0.0392 − 0.146i)14-s + (0.234 + 0.0855i)16-s + (−0.928 + 1.32i)17-s + (0.132 − 0.0115i)19-s + (0.273 + 0.191i)20-s + (−0.799 + 0.372i)22-s + (−1.86 + 0.499i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.984 + 0.175i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.984 + 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0349212 - 0.394591i\)
\(L(\frac12)\) \(\approx\) \(0.0349212 - 0.394591i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.73 + 2.03i)T \)
good5 \( 1 + (1.35 + 0.630i)T + (3.21 + 3.83i)T^{2} \)
7 \( 1 + (0.533 - 0.194i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.06 - 3.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.18 + 1.69i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (3.83 - 5.46i)T + (-5.81 - 15.9i)T^{2} \)
19 \( 1 + (-0.576 + 0.0503i)T + (18.7 - 3.29i)T^{2} \)
23 \( 1 + (8.94 - 2.39i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.66 + 2.05i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (2.71 + 2.71i)T + 31iT^{2} \)
41 \( 1 + (0.166 - 0.945i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (6.19 - 6.19i)T - 43iT^{2} \)
47 \( 1 + (-7.18 - 4.14i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.62 - 4.47i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.589 + 1.26i)T + (-37.9 + 45.1i)T^{2} \)
61 \( 1 + (-0.953 + 0.667i)T + (20.8 - 57.3i)T^{2} \)
67 \( 1 + (-1.02 - 2.82i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-8.44 + 10.0i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 - 2.55iT - 73T^{2} \)
79 \( 1 + (-0.0758 + 0.162i)T + (-50.7 - 60.5i)T^{2} \)
83 \( 1 + (-8.12 + 1.43i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (10.1 - 4.71i)T + (57.2 - 68.1i)T^{2} \)
97 \( 1 + (-4.90 - 18.3i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.93200977782930688476858844136, −9.810091542177419538525162901557, −9.292347903216091800317580701648, −8.043661805619472146710730324045, −7.69799560088306949066716636978, −6.50432141596708952800902677100, −5.76743026310236801674644820999, −4.40751250479832717286931930876, −3.88064257562061133740269092732, −1.95404287521821461214952276767, 0.20750995981882649835525238815, 2.08973976993259215374373240846, 3.41043421255887661276257656189, 4.12098241897561066897259684782, 5.37990639142315267452674119862, 6.57080425444565207117325382640, 7.47609485317884922696783182280, 8.519453172284726081295329219808, 9.276049870922640515949152884754, 10.08207915899300221824158059582

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