Properties

Label 2-666-1.1-c1-0-5
Degree $2$
Conductor $666$
Sign $1$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.30·5-s + 4.60·7-s + 8-s − 1.30·10-s − 1.30·11-s − 2.30·13-s + 4.60·14-s + 16-s + 6·17-s + 2·19-s − 1.30·20-s − 1.30·22-s + 6.90·23-s − 3.30·25-s − 2.30·26-s + 4.60·28-s − 6.90·29-s + 3.30·31-s + 32-s + 6·34-s − 6·35-s + 37-s + 2·38-s − 1.30·40-s + 0.908·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.582·5-s + 1.74·7-s + 0.353·8-s − 0.411·10-s − 0.392·11-s − 0.638·13-s + 1.23·14-s + 0.250·16-s + 1.45·17-s + 0.458·19-s − 0.291·20-s − 0.277·22-s + 1.44·23-s − 0.660·25-s − 0.451·26-s + 0.870·28-s − 1.28·29-s + 0.593·31-s + 0.176·32-s + 1.02·34-s − 1.01·35-s + 0.164·37-s + 0.324·38-s − 0.205·40-s + 0.141·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.475317026\)
\(L(\frac12)\) \(\approx\) \(2.475317026\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
37 \( 1 - T \)
good5 \( 1 + 1.30T + 5T^{2} \)
7 \( 1 - 4.60T + 7T^{2} \)
11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 6.90T + 23T^{2} \)
29 \( 1 + 6.90T + 29T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
41 \( 1 - 0.908T + 41T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 - 2.60T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 3.39T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 8.69T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + 17.2T + 83T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.80173622629491757159567795677, −9.816211648176531494013209199008, −8.522826508070270413003441427636, −7.66593324608740923429045774346, −7.28619872223283029369478003157, −5.61225369719099592638107667636, −5.05431496778729029187448796252, −4.12270267488747912083793452666, −2.89482296470876112563179558494, −1.45791910636920151145052736404, 1.45791910636920151145052736404, 2.89482296470876112563179558494, 4.12270267488747912083793452666, 5.05431496778729029187448796252, 5.61225369719099592638107667636, 7.28619872223283029369478003157, 7.66593324608740923429045774346, 8.522826508070270413003441427636, 9.816211648176531494013209199008, 10.80173622629491757159567795677

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