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Properties

Label 2-666-111.98-c1-0-0
Degree $2$
Conductor $666$
Sign $-0.645 - 0.763i$
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 + (1.31 + 0.114i)5-s + (−1.79 + 1.51i)7-s + (0.965 − 0.258i)8-s + (−0.658 − 1.14i)10-s + (−1.26 + 2.19i)11-s + (−5.68 + 2.65i)13-s + (2.26 + 0.608i)14-s + (−0.766 − 0.642i)16-s + (−2.17 − 1.01i)17-s + (−5.10 − 3.57i)19-s + (−0.556 + 1.19i)20-s + (2.52 − 0.221i)22-s + (1.95 − 7.30i)23-s + ⋯
L(s)  = 1  + (−0.405 − 0.579i)2-s + (−0.171 + 0.469i)4-s + (0.587 + 0.0513i)5-s + (−0.680 + 0.570i)7-s + (0.341 − 0.0915i)8-s + (−0.208 − 0.360i)10-s + (−0.382 + 0.662i)11-s + (−1.57 + 0.735i)13-s + (0.606 + 0.162i)14-s + (−0.191 − 0.160i)16-s + (−0.526 − 0.245i)17-s + (−1.17 − 0.820i)19-s + (−0.124 + 0.267i)20-s + (0.538 − 0.0471i)22-s + (0.408 − 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.645 - 0.763i$
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: \((2,\ 666,\ (\ :1/2),\ -0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124564 + 0.268276i\)
\(L(\frac12)\) \(\approx\) \(0.124564 + 0.268276i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.02 + 0.866i)T \)
good5 \( 1 + (-1.31 - 0.114i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (1.79 - 1.51i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (1.26 - 2.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.68 - 2.65i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (2.17 + 1.01i)T + (10.9 + 13.0i)T^{2} \)
19 \( 1 + (5.10 + 3.57i)T + (6.49 + 17.8i)T^{2} \)
23 \( 1 + (-1.95 + 7.30i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.27 - 8.48i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.77 - 1.77i)T + 31iT^{2} \)
41 \( 1 + (10.3 + 3.74i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.789 - 0.789i)T - 43iT^{2} \)
47 \( 1 + (-9.67 + 5.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.18 - 1.41i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-0.761 - 8.70i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (-4.69 - 10.0i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (-5.02 - 5.99i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (4.03 - 0.711i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + (-0.818 + 9.35i)T + (-77.7 - 13.7i)T^{2} \)
83 \( 1 + (-0.388 - 1.06i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.317 + 0.0277i)T + (87.6 - 15.4i)T^{2} \)
97 \( 1 + (-12.8 - 3.43i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.41996433493717024790763360078, −10.18415232992137739439415636357, −9.038172596833064552719142793423, −8.729876948629715414314125475970, −7.11277555971713332823859402691, −6.69676181866716138015028267886, −5.21892978423689253660983587730, −4.37779694985330356764197615045, −2.69635650782155329414000392666, −2.15521313097231060019912821253, 0.16288020257869580931322319848, 2.11723774832707052984414094631, 3.55259856942433448798839150086, 4.92067928385727593669315550712, 5.85313489028721498177464327951, 6.62391085058109591956051028421, 7.64368442824694157496259695484, 8.314274491817895999694188738325, 9.562329293869585571755482135741, 9.973845834616758908182545601084

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