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Properties

Label 2-666-37.16-c1-0-6
Degree $2$
Conductor $666$
Sign $0.527 - 0.849i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.326 − 1.85i)5-s + (−0.711 + 4.03i)7-s + (0.500 + 0.866i)8-s + (0.939 − 1.62i)10-s + (1.67 + 2.89i)11-s + (1.48 + 1.24i)13-s + (−2.04 + 3.55i)14-s + (0.173 + 0.984i)16-s + (−3.40 + 2.85i)17-s + (0.0932 − 0.0339i)19-s + (1.43 − 1.20i)20-s + (0.580 + 3.29i)22-s + (4.76 − 8.24i)23-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (0.145 − 0.827i)5-s + (−0.269 + 1.52i)7-s + (0.176 + 0.306i)8-s + (0.297 − 0.514i)10-s + (0.503 + 0.872i)11-s + (0.411 + 0.345i)13-s + (−0.547 + 0.948i)14-s + (0.0434 + 0.246i)16-s + (−0.826 + 0.693i)17-s + (0.0213 − 0.00778i)19-s + (0.321 − 0.270i)20-s + (0.123 + 0.701i)22-s + (0.992 − 1.71i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.527 - 0.849i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95710 + 1.08811i\)
\(L(\frac12)\) \(\approx\) \(1.95710 + 1.08811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 \)
37 \( 1 + (3.50 - 4.97i)T \)
good5 \( 1 + (-0.326 + 1.85i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.711 - 4.03i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-1.67 - 2.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.48 - 1.24i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (3.40 - 2.85i)T + (2.95 - 16.7i)T^{2} \)
19 \( 1 + (-0.0932 + 0.0339i)T + (14.5 - 12.2i)T^{2} \)
23 \( 1 + (-4.76 + 8.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.387 - 0.670i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
41 \( 1 + (4.36 + 3.66i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + (3.55 - 6.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.142 + 0.806i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.40 + 7.97i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (8.19 + 6.87i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.28 + 12.9i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.79 - 1.01i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + 2.55T + 73T^{2} \)
79 \( 1 + (0.943 - 5.35i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.504 - 0.423i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (0.612 + 3.47i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.81 - 3.13i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.79738289203314036383972370110, −9.590548099683586945074457302953, −8.756385989398218096258083460052, −8.336062415715434325897268237979, −6.67228691620217828317489279650, −6.28323813848785324754089599806, −4.99996471551196810114014435884, −4.51300129750981284462623850526, −2.96587343376994681652234832587, −1.80641380633240933061714379314, 1.07510387842674868010768773128, 2.94231431208544997787617027697, 3.62578949316181445674247134369, 4.68282446963819053278327007279, 5.96019031951350329626555188925, 6.83169675189239457364222632765, 7.35013227583370331654224120787, 8.682272448768393196992692808958, 9.854127480265660345410884126637, 10.55059306737152198703615677386

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