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Properties

Label 2-666-37.33-c1-0-6
Degree $2$
Conductor $666$
Sign $0.267 - 0.963i$
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 + (−0.365 + 0.307i)7-s + (0.5 − 0.866i)8-s + (−0.173 − 0.300i)10-s + (1.29 − 2.23i)11-s + (4.21 + 1.53i)13-s + (−0.238 − 0.413i)14-s + (0.766 + 0.642i)16-s + (1.88 − 0.687i)17-s + (0.611 + 3.46i)19-s + (0.326 − 0.118i)20-s + (1.98 + 1.66i)22-s + (2.60 + 4.51i)23-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.469 − 0.171i)4-s + (−0.118 + 0.0998i)5-s + (−0.138 + 0.116i)7-s + (0.176 − 0.306i)8-s + (−0.0549 − 0.0951i)10-s + (0.389 − 0.675i)11-s + (1.16 + 0.425i)13-s + (−0.0638 − 0.110i)14-s + (0.191 + 0.160i)16-s + (0.458 − 0.166i)17-s + (0.140 + 0.795i)19-s + (0.0729 − 0.0265i)20-s + (0.422 + 0.354i)22-s + (0.543 + 0.940i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.267 - 0.963i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07991 + 0.820845i\)
\(L(\frac12)\) \(\approx\) \(1.07991 + 0.820845i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 + 5.57i)T \)
good5 \( 1 + (0.266 - 0.223i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.365 - 0.307i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.29 + 2.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.21 - 1.53i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.88 + 0.687i)T + (13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.611 - 3.46i)T + (-17.8 + 6.49i)T^{2} \)
23 \( 1 + (-2.60 - 4.51i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.114 + 0.197i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
41 \( 1 + (-10.0 - 3.66i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 - 8.85T + 43T^{2} \)
47 \( 1 + (-3.72 - 6.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 + 2.81i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.43 + 2.87i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.35 - 0.491i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (8.06 - 6.76i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.54 + 8.74i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 - 7.18T + 73T^{2} \)
79 \( 1 + (-2.70 + 2.27i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.32 + 0.483i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.92 + 2.45i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (9.24 + 16.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.85708769342812982089358741670, −9.379811412320306757932671619116, −9.131259077560178742645880272244, −7.931300328332695915471087923169, −7.29567560958031821858299981411, −6.05670011650124538300386262913, −5.66086691712369149798513234652, −4.15926916985988573999799094717, −3.29281739341288898064724198233, −1.30604052159233672772822479720, 0.938231514004967941968975425472, 2.45168316149277629546052981935, 3.67901042615026390786623162344, 4.54977124947371554223672036827, 5.76086094220957390080115011486, 6.83817029408724175680789531320, 7.88246074775791512356323283474, 8.799502569067049998992593452363, 9.482960335499094624091734776380, 10.53061396126948863343109058190

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