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Properties

Label 2-666-37.36-c1-0-0
Degree $2$
Conductor $666$
Sign $-0.753 + 0.657i$
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  + i·2-s − 4-s + 3.79i·5-s − 2·7-s − i·8-s − 3.79·10-s − 3.79·11-s + 0.791i·13-s − 2i·14-s + 16-s + 1.58i·17-s − 7.58i·19-s − 3.79i·20-s − 3.79i·22-s − 0.791i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.69i·5-s − 0.755·7-s − 0.353i·8-s − 1.19·10-s − 1.14·11-s + 0.219i·13-s − 0.534i·14-s + 0.250·16-s + 0.383i·17-s − 1.73i·19-s − 0.847i·20-s − 0.808i·22-s − 0.164i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190086 - 0.506834i\)
\(L(\frac12)\) \(\approx\) \(0.190086 - 0.506834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
37 \( 1 + (-4 - 4.58i)T \)
good5 \( 1 - 3.79iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 - 0.791iT - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 + 7.58iT - 19T^{2} \)
23 \( 1 + 0.791iT - 23T^{2} \)
29 \( 1 - 0.791iT - 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
41 \( 1 + 5.20T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 + 1.58T + 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 - 8.20iT - 61T^{2} \)
67 \( 1 + 7.37T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 4.41iT - 97T^{2} \)
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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

−10.75488256709262226526661169343, −10.28929436318678922861526341096, −9.354588881532662324594902557084, −8.276855128813773325338427107034, −7.21134759533097361410516820749, −6.79052660820167491008033271997, −5.95197524910413501191930560600, −4.77260010823549206405443747235, −3.34420486172461216336497628818, −2.60766675753457366902952039985, 0.27756468814704089818912181389, 1.76939272541796559300760073266, 3.23479268065133357380490257308, 4.36858364394398006052302008151, 5.26826617512832512140687737241, 6.05326525230033143737409132792, 7.82562933600375919644057694482, 8.240807796249019618213388025329, 9.449638943512191518881329597306, 9.759890212200888989125797503807

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