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Properties

Label 2-666-333.175-c1-0-31
Degree $2$
Conductor $666$
Sign $-0.985 + 0.169i$
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  − i·2-s + (0.0965 + 1.72i)3-s − 4-s − 2.20i·5-s + (1.72 − 0.0965i)6-s + (−0.875 + 1.51i)7-s + i·8-s + (−2.98 + 0.333i)9-s − 2.20·10-s + (−2.89 − 5.01i)11-s + (−0.0965 − 1.72i)12-s + 1.73i·13-s + (1.51 + 0.875i)14-s + (3.80 − 0.212i)15-s + 16-s + (−1.78 − 1.02i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.0557 + 0.998i)3-s − 0.5·4-s − 0.985i·5-s + (0.706 − 0.0394i)6-s + (−0.331 + 0.573i)7-s + 0.353i·8-s + (−0.993 + 0.111i)9-s − 0.696·10-s + (−0.872 − 1.51i)11-s + (−0.0278 − 0.499i)12-s + 0.480i·13-s + (0.405 + 0.234i)14-s + (0.983 − 0.0549i)15-s + 0.250·16-s + (−0.432 − 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0323271 - 0.377663i\)
\(L(\frac12)\) \(\approx\) \(0.0323271 - 0.377663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.0965 - 1.72i)T \)
37 \( 1 + (-2.05 + 5.72i)T \)
good5 \( 1 + 2.20iT - 5T^{2} \)
7 \( 1 + (0.875 - 1.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.89 + 5.01i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (1.78 + 1.02i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.49 - 1.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.24 + 2.44i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.02 - 1.74i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.00 + 2.89i)T + (15.5 + 26.8i)T^{2} \)
41 \( 1 + 0.153T + 41T^{2} \)
43 \( 1 + (1.67 - 0.965i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.21 + 3.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.36 + 5.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.26 - 1.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.05 + 2.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + (-4.20 - 7.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.77T + 73T^{2} \)
79 \( 1 + (-8.51 - 4.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 + (-12.2 - 7.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 6.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.17793476470895531675832554545, −9.090372057889463211557270784594, −8.833577442641869559377326229722, −7.977416879925682432718158856970, −6.04263350375831216331481374193, −5.39011473029950713431258382807, −4.41749817888539674265320535709, −3.45327881641020268759907274155, −2.30355343042606266146583670612, −0.18679632736011610915201641219, 2.02883135074157593828643056077, 3.23730246208598950499134085012, 4.63160374450156005301114233515, 5.88241740214936824818668345008, 6.70594117181056361948341002938, 7.38720523753283789334054270976, 7.83187422688554806748433750120, 9.046329676288216763385064310550, 10.16744190889074894052227845726, 10.71545832900226888710623700629

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