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Properties

Label 2-666-111.17-c1-0-6
Degree $2$
Conductor $666$
Sign $0.604 - 0.796i$
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 + (3.55 − 0.310i)5-s + (3.11 + 2.61i)7-s + (0.965 + 0.258i)8-s + (−1.78 + 3.08i)10-s + (−0.00168 − 0.00292i)11-s + (1.48 + 0.692i)13-s + (−3.92 + 1.05i)14-s + (−0.766 + 0.642i)16-s + (−6.72 + 3.13i)17-s + (5.40 − 3.78i)19-s + (−1.50 − 3.23i)20-s + (0.00335 + 0.000293i)22-s + (−1.87 − 7.01i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (1.58 − 0.138i)5-s + (1.17 + 0.987i)7-s + (0.341 + 0.0915i)8-s + (−0.563 + 0.976i)10-s + (−0.000508 − 0.000880i)11-s + (0.411 + 0.192i)13-s + (−1.04 + 0.281i)14-s + (−0.191 + 0.160i)16-s + (−1.62 + 0.760i)17-s + (1.24 − 0.868i)19-s + (−0.336 − 0.722i)20-s + (0.000716 + 6.26e−5i)22-s + (−0.391 − 1.46i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.604 - 0.796i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57618 + 0.782060i\)
\(L(\frac12)\) \(\approx\) \(1.57618 + 0.782060i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.96 + 1.17i)T \)
good5 \( 1 + (-3.55 + 0.310i)T + (4.92 - 0.868i)T^{2} \)
7 \( 1 + (-3.11 - 2.61i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.00168 + 0.00292i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.48 - 0.692i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (6.72 - 3.13i)T + (10.9 - 13.0i)T^{2} \)
19 \( 1 + (-5.40 + 3.78i)T + (6.49 - 17.8i)T^{2} \)
23 \( 1 + (1.87 + 7.01i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.451 + 1.68i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.34 - 6.34i)T - 31iT^{2} \)
41 \( 1 + (8.09 - 2.94i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.92 + 3.92i)T + 43iT^{2} \)
47 \( 1 + (-0.830 - 0.479i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.38 - 8.80i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-0.918 + 10.4i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (4.66 - 10.0i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (-6.84 + 8.15i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (6.56 + 1.15i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 - 1.04iT - 73T^{2} \)
79 \( 1 + (-0.482 - 5.51i)T + (-77.7 + 13.7i)T^{2} \)
83 \( 1 + (0.623 - 1.71i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (-3.68 - 0.322i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (4.41 - 1.18i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.57418565574738507736876536882, −9.523883991506982607859041747288, −8.812557352648763362421677493967, −8.393105638659608257067840246812, −6.95451118930837819917732903005, −6.14795279714029261345233328330, −5.36463154055608813778803278783, −4.61369635983886884646666836768, −2.43528158632527716350426814694, −1.60339078156321831662609827736, 1.33119681068941289755539490713, 2.16025199402190346911856046325, 3.65331563545021361257802836970, 4.89671110800636781151159922479, 5.77457167000576629906680353412, 7.02179621023483124388654621295, 7.81397691307231715227903780406, 8.905857876144214569354396725861, 9.706440157704821836141107902373, 10.28213042592635947888343937092

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