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Properties

Label 2-666-37.26-c1-0-12
Degree $2$
Conductor $666$
Sign $-0.227 + 0.973i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.5 − 2.59i)5-s + (2 − 3.46i)7-s + 0.999·8-s − 3·10-s + 6·11-s + (−1 + 1.73i)13-s − 3.99·14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s + (−1 + 1.73i)19-s + (1.50 + 2.59i)20-s + (−3 − 5.19i)22-s − 6·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + (0.755 − 1.30i)7-s + 0.353·8-s − 0.948·10-s + 1.80·11-s + (−0.277 + 0.480i)13-s − 1.06·14-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.229 + 0.397i)19-s + (0.335 + 0.580i)20-s + (−0.639 − 1.10i)22-s − 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980902 - 1.23633i\)
\(L(\frac12)\) \(\approx\) \(0.980902 - 1.23633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
37 \( 1 + (-5.5 + 2.59i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−10.10902638899157411483228433622, −9.521764319562659199178245474575, −8.690173801416819714488735108548, −7.901327502592385876726466737549, −6.81126569717800870079943760346, −5.65873858784129263721906016563, −4.27080625255500743610972926774, −4.00965329015128449021539138435, −1.79613906679756125345681931643, −1.12471837199076816930995517139, 1.75341270264198906883547249232, 2.90142315331319424496889821231, 4.47032060414079140953111468613, 5.76002121126875637677185898293, 6.22607661180026523194009299268, 7.17025985439968865239515539619, 8.157508080426453968141298082709, 9.127107215105970828885280331669, 9.664678436761957527124219106579, 10.64307399443433976726192149390

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