Properties

Label 2-666-1.1-c1-0-1
Degree $2$
Conductor $666$
Sign $1$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 3·7-s − 8-s − 2·10-s + 5·11-s − 3·13-s + 3·14-s + 16-s + 3·17-s + 5·19-s + 2·20-s − 5·22-s + 3·23-s − 25-s + 3·26-s − 3·28-s + 4·31-s − 32-s − 3·34-s − 6·35-s + 37-s − 5·38-s − 2·40-s + 6·41-s + 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.13·7-s − 0.353·8-s − 0.632·10-s + 1.50·11-s − 0.832·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 0.447·20-s − 1.06·22-s + 0.625·23-s − 1/5·25-s + 0.588·26-s − 0.566·28-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.01·35-s + 0.164·37-s − 0.811·38-s − 0.316·40-s + 0.937·41-s + 0.609·43-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: yes
Arithmetic: yes
Character: $\chi_{666} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.223119248\)
\(L(\frac12)\) \(\approx\) \(1.223119248\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−10.11202593962156360987559406213, −9.479645883044483591577807946944, −9.315757253980273759687218979884, −7.913878554875197130364198555006, −6.87960233611251053044806376508, −6.29485884015171456248050146449, −5.29912923560305098381281180294, −3.69526832182271691380022426077, −2.59442247256705920291581006864, −1.12099772831246061899779950887, 1.12099772831246061899779950887, 2.59442247256705920291581006864, 3.69526832182271691380022426077, 5.29912923560305098381281180294, 6.29485884015171456248050146449, 6.87960233611251053044806376508, 7.913878554875197130364198555006, 9.315757253980273759687218979884, 9.479645883044483591577807946944, 10.11202593962156360987559406213

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