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Properties

Label 2-666-37.9-c1-0-15
Degree $2$
Conductor $666$
Sign $-0.993 + 0.116i$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.266 − 0.223i)5-s + (0.773 + 0.649i)7-s + (0.5 + 0.866i)8-s + (−0.173 + 0.300i)10-s + (−2.73 − 4.73i)11-s + (−5.97 + 2.17i)13-s + (0.504 − 0.874i)14-s + (0.766 − 0.642i)16-s + (0.490 + 0.178i)17-s + (0.869 − 4.93i)19-s + (0.326 + 0.118i)20-s + (−4.18 + 3.51i)22-s + (0.721 − 1.24i)23-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (−0.118 − 0.0998i)5-s + (0.292 + 0.245i)7-s + (0.176 + 0.306i)8-s + (−0.0549 + 0.0951i)10-s + (−0.823 − 1.42i)11-s + (−1.65 + 0.603i)13-s + (0.134 − 0.233i)14-s + (0.191 − 0.160i)16-s + (0.119 + 0.0433i)17-s + (0.199 − 1.13i)19-s + (0.0729 + 0.0265i)20-s + (−0.892 + 0.748i)22-s + (0.150 − 0.260i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0363591 - 0.622208i\)
\(L(\frac12)\) \(\approx\) \(0.0363591 - 0.622208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 \)
37 \( 1 + (5.88 - 1.53i)T \)
good5 \( 1 + (0.266 + 0.223i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.773 - 0.649i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (2.73 + 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.97 - 2.17i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.490 - 0.178i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-0.869 + 4.93i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-0.721 + 1.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.16 + 5.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
41 \( 1 + (1.03 - 0.376i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + (5.71 - 9.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0706 - 0.0593i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (2.29 - 1.92i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (3.58 - 1.30i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.892 + 0.748i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.0543 - 0.308i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + (-7.12 - 5.98i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.89 - 3.23i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-12.0 + 10.1i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.464 + 0.804i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.13556759166392988609182837037, −9.354983001861693518880338574403, −8.422914127235538523452172949660, −7.74765096172053458332467390439, −6.53229336041695525501882289261, −5.26180378173184383738691248700, −4.56637308583442742776657776057, −3.12588525505060107679535247726, −2.24094727434443105193779167700, −0.32879356204538299275772602722, 1.93423725160281862749324298382, 3.47605580496576728694807373637, 4.95540311898102083933506099915, 5.23111670896196396336389331537, 6.76143807509163166107939383912, 7.56840114755383099172095847984, 7.904278651939843318347126154078, 9.287759703574053876117740828011, 10.02523052701226341003352747413, 10.58154902973708656803735098506

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