Properties

Label 2-666-37.21-c1-0-4
Degree $2$
Conductor $666$
Sign $0.992 - 0.119i$
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  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (−2.57 − 0.453i)5-s + (−0.361 + 2.04i)7-s + (−0.866 + 0.500i)8-s + (−1.30 + 2.26i)10-s + (2.99 + 5.19i)11-s + (2.64 − 3.15i)13-s + (1.80 + 1.03i)14-s + (0.173 + 0.984i)16-s + (0.618 + 0.737i)17-s + (0.534 + 1.46i)19-s + (1.67 + 2.00i)20-s + (5.90 − 1.04i)22-s + (5.51 + 3.18i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (−0.383 − 0.321i)4-s + (−1.15 − 0.202i)5-s + (−0.136 + 0.773i)7-s + (−0.306 + 0.176i)8-s + (−0.412 + 0.715i)10-s + (0.903 + 1.56i)11-s + (0.733 − 0.874i)13-s + (0.481 + 0.277i)14-s + (0.0434 + 0.246i)16-s + (0.150 + 0.178i)17-s + (0.122 + 0.337i)19-s + (0.375 + 0.447i)20-s + (1.25 − 0.221i)22-s + (1.15 + 0.664i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.992 - 0.119i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.992 - 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29109 + 0.0774198i\)
\(L(\frac12)\) \(\approx\) \(1.29109 + 0.0774198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 \)
37 \( 1 + (-6.07 - 0.383i)T \)
good5 \( 1 + (2.57 + 0.453i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.361 - 2.04i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.99 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.64 + 3.15i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.618 - 0.737i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-0.534 - 1.46i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (-5.51 - 3.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 + 2.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.39iT - 31T^{2} \)
41 \( 1 + (7.94 + 6.66i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 - 3.76iT - 43T^{2} \)
47 \( 1 + (3.08 - 5.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.39 - 7.90i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (5.02 - 0.885i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (6.25 - 7.45i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.83 - 10.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-10.1 + 3.69i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + (-2.51 - 0.442i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.29 - 4.44i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-16.0 + 2.82i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (14.1 + 8.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.66692036805182079948609540685, −9.692738999890937431382216582739, −8.915080084945840783816892263320, −8.033514700084843602037112005585, −7.08173031551864704038390206210, −5.88771093609375291460271995346, −4.77260785964171383169928342979, −3.92896580894098711942845467220, −2.91814945480366620069647992350, −1.34881592513410633704648068195, 0.76215683280277522638336837626, 3.32825285484429288238902194958, 3.86873384581332430810741349790, 4.90655393075669276084180733574, 6.38414372882331104285269468053, 6.78030588507744196525647179096, 7.87423986300902971962495076634, 8.568532094079508919659968136548, 9.358290872636202746994609401756, 10.76428291104762291267928751982

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