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Properties

Label 2-666-111.56-c1-0-8
Degree $2$
Conductor $666$
Sign $0.793 + 0.609i$
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 + (−3.03 + 1.41i)5-s + (−1.63 − 0.593i)7-s + (−0.258 − 0.965i)8-s + (−1.67 − 2.90i)10-s + (0.408 − 0.708i)11-s + (3.82 − 5.46i)13-s + (0.449 − 1.67i)14-s + (0.939 − 0.342i)16-s + (−0.771 − 1.10i)17-s + (−1.63 − 0.143i)19-s + (2.74 − 1.92i)20-s + (0.741 + 0.345i)22-s + (4.85 + 1.30i)23-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (−0.492 + 0.0868i)4-s + (−1.35 + 0.633i)5-s + (−0.616 − 0.224i)7-s + (−0.0915 − 0.341i)8-s + (−0.529 − 0.917i)10-s + (0.123 − 0.213i)11-s + (1.06 − 1.51i)13-s + (0.120 − 0.447i)14-s + (0.234 − 0.0855i)16-s + (−0.187 − 0.267i)17-s + (−0.375 − 0.0328i)19-s + (0.613 − 0.429i)20-s + (0.158 + 0.0736i)22-s + (1.01 + 0.271i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.793 + 0.609i$
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: \((2,\ 666,\ (\ :1/2),\ 0.793 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692346 - 0.235240i\)
\(L(\frac12)\) \(\approx\) \(0.692346 - 0.235240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.15 - 4.44i)T \)
good5 \( 1 + (3.03 - 1.41i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (1.63 + 0.593i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.408 + 0.708i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.82 + 5.46i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (0.771 + 1.10i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (1.63 + 0.143i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (-4.85 - 1.30i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.12 - 0.836i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-5.52 + 5.52i)T - 31iT^{2} \)
41 \( 1 + (0.631 + 3.58i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.38 + 7.38i)T + 43iT^{2} \)
47 \( 1 + (2.69 - 1.55i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.64 + 4.52i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-3.21 + 6.88i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (9.61 + 6.73i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (-1.99 + 5.49i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (7.41 + 8.83i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + 6.79iT - 73T^{2} \)
79 \( 1 + (-3.68 - 7.89i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (-8.80 - 1.55i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (4.08 + 1.90i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (2.31 - 8.63i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.55654072460096924278946238585, −9.459690649792886892853163294730, −8.311175144417039071428391503374, −7.86177788437599141743705070693, −6.88136915696703809660136602134, −6.18283122525973951605841583670, −4.95890576675275804469076101246, −3.68454885456507905286486386586, −3.18422521082688788867308286986, −0.43049277759133843827155740965, 1.34685332882723795604818224346, 3.05007891741258367612622883405, 4.10456004055660333727060006002, 4.63203312481063445882096467510, 6.15884312984330382919257187675, 7.10285527230391148202890227951, 8.348726023114897360418677680308, 8.847580127483380161942492587755, 9.669646127144300892904142293545, 10.89907079499436518003660856464

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