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Properties

Label 2-666-111.98-c1-0-12
Degree $2$
Conductor $666$
Sign $0.265 + 0.964i$
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  + (0.573 + 0.819i)2-s + (−0.342 + 0.939i)4-s + (−3.66 − 0.320i)5-s + (−1.19 + 0.999i)7-s + (−0.965 + 0.258i)8-s + (−1.83 − 3.18i)10-s + (1.53 − 2.65i)11-s + (4.68 − 2.18i)13-s + (−1.50 − 0.402i)14-s + (−0.766 − 0.642i)16-s + (−3.89 − 1.81i)17-s + (−3.65 − 2.55i)19-s + (1.55 − 3.33i)20-s + (3.05 − 0.267i)22-s + (1.44 − 5.38i)23-s + ⋯
L(s)  = 1  + (0.405 + 0.579i)2-s + (−0.171 + 0.469i)4-s + (−1.63 − 0.143i)5-s + (−0.450 + 0.377i)7-s + (−0.341 + 0.0915i)8-s + (−0.581 − 1.00i)10-s + (0.462 − 0.801i)11-s + (1.29 − 0.605i)13-s + (−0.401 − 0.107i)14-s + (−0.191 − 0.160i)16-s + (−0.945 − 0.440i)17-s + (−0.837 − 0.586i)19-s + (0.347 − 0.744i)20-s + (0.651 − 0.0570i)22-s + (0.300 − 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.265 + 0.964i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557109 - 0.424610i\)
\(L(\frac12)\) \(\approx\) \(0.557109 - 0.424610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.573 - 0.819i)T \)
3 \( 1 \)
37 \( 1 + (5.10 + 3.31i)T \)
good5 \( 1 + (3.66 + 0.320i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (1.19 - 0.999i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.53 + 2.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.68 + 2.18i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (3.89 + 1.81i)T + (10.9 + 13.0i)T^{2} \)
19 \( 1 + (3.65 + 2.55i)T + (6.49 + 17.8i)T^{2} \)
23 \( 1 + (-1.44 + 5.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.43 + 5.35i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.808 - 0.808i)T + 31iT^{2} \)
41 \( 1 + (4.40 + 1.60i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.81 + 2.81i)T - 43iT^{2} \)
47 \( 1 + (6.67 - 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.78 - 2.12i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-0.929 - 10.6i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (2.73 + 5.86i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (-6.26 - 7.46i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (11.1 - 1.96i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + 2.09iT - 73T^{2} \)
79 \( 1 + (-1.21 + 13.8i)T + (-77.7 - 13.7i)T^{2} \)
83 \( 1 + (2.26 + 6.23i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-14.6 + 1.28i)T + (87.6 - 15.4i)T^{2} \)
97 \( 1 + (14.4 + 3.88i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.64962786519028941311719859693, −8.810259852736112819754800201473, −8.724081585318781348737570135543, −7.72049989634397983902838252928, −6.70262286894103959514628691243, −6.00867885663490154688773003147, −4.64192255842873129245213693433, −3.87137283171039491305086873754, −2.96197543928440508210099601630, −0.33695527506854071193696059321, 1.62666793621757949924902992525, 3.51441091780953809708412053834, 3.86542591107420790973095992450, 4.82378878020773330176858287130, 6.43154005869550136077081712146, 7.01267899423937689385368309989, 8.195107761778776858404760843338, 8.929744288781685510495790535892, 10.03756460702319004413019172353, 11.04025842205276594166082861424

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