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Properties

Label 2-666-37.11-c1-0-6
Degree $2$
Conductor $666$
Sign $0.367 + 0.929i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + (0.633 − 1.09i)7-s + 0.999i·8-s + 0.999·10-s − 4.73·11-s + (4.73 + 2.73i)13-s + 1.26i·14-s + (−0.5 − 0.866i)16-s + (6.69 − 3.86i)17-s + (−5.83 − 3.36i)19-s + (−0.866 + 0.499i)20-s + (4.09 − 2.36i)22-s − 1.46i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.239 − 0.415i)7-s + 0.353i·8-s + 0.316·10-s − 1.42·11-s + (1.31 + 0.757i)13-s + 0.338i·14-s + (−0.125 − 0.216i)16-s + (1.62 − 0.937i)17-s + (−1.33 − 0.772i)19-s + (−0.193 + 0.111i)20-s + (0.873 − 0.504i)22-s − 0.305i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687787 - 0.467668i\)
\(L(\frac12)\) \(\approx\) \(0.687787 - 0.467668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
37 \( 1 + (6.06 - 0.5i)T \)
good5 \( 1 + (0.866 + 0.5i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.633 + 1.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.69 + 3.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.83 + 3.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 0.464iT - 29T^{2} \)
31 \( 1 + 6.19iT - 31T^{2} \)
41 \( 1 + (-4.23 + 7.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 5.26iT - 43T^{2} \)
47 \( 1 - 5.26T + 47T^{2} \)
53 \( 1 + (4.46 + 7.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 3.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.33 + 4.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.83 + 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.63 - 4.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (-5.36 - 3.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.26 + 5.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 4.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.33iT - 97T^{2} \)
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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

−10.44564126795321333798997209030, −9.393481573054499433930520211212, −8.458777884376278679024243775286, −7.86520830918177183790695438092, −7.01680163453890182028361157901, −5.93711951590814295462850842329, −4.95057923473085395294709247875, −3.81357603511856991176926327972, −2.28407665196525760525246953201, −0.57061564323279276290378339986, 1.48086640914669884695660055302, 2.95859953069710348367116392654, 3.82567464463763001301904548577, 5.40325040303042630695565998647, 6.14259278086528115312273468744, 7.62488358279123191196828655305, 8.095658651653836932364881818490, 8.764961462413206188894536689844, 10.13241743961508650058624511859, 10.56283540594214863763072051322

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