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Properties

Label 2-666-37.36-c1-0-0
Degree 22
Conductor 666666
Sign −0.753+0.657i-0.753 + 0.657i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(666s/2 ΓC(s) L(s)=((−0.753+0.657i) Λ‾(2−s)\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}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((−0.753+0.657i) Λ‾(1−s)\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: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −0.753+0.657i-0.753 + 0.657i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(73,⋅)\chi_{666} (73, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), −0.753+0.657i)(2,\ 666,\ (\ :1/2),\ -0.753 + 0.657i)

Particular Values

L(1)L(1) ≈\approx 0.190086−0.506834i0.190086 - 0.506834i
L(12)L(\frac12) ≈\approx 0.190086−0.506834i0.190086 - 0.506834i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−iT 1 - iT
3 1 1
37 1+(−4−4.58i)T 1 + (-4 - 4.58i)T
good5 1−3.79iT−5T2 1 - 3.79iT - 5T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+3.79T+11T2 1 + 3.79T + 11T^{2}
13 1−0.791iT−13T2 1 - 0.791iT - 13T^{2}
17 1−1.58iT−17T2 1 - 1.58iT - 17T^{2}
19 1+7.58iT−19T2 1 + 7.58iT - 19T^{2}
23 1+0.791iT−23T2 1 + 0.791iT - 23T^{2}
29 1−0.791iT−29T2 1 - 0.791iT - 29T^{2}
31 1−5.37iT−31T2 1 - 5.37iT - 31T^{2}
41 1+5.20T+41T2 1 + 5.20T + 41T^{2}
43 1+6iT−43T2 1 + 6iT - 43T^{2}
47 1+1.58T+47T2 1 + 1.58T + 47T^{2}
53 1+7.58T+53T2 1 + 7.58T + 53T^{2}
59 1−7.58iT−59T2 1 - 7.58iT - 59T^{2}
61 1−8.20iT−61T2 1 - 8.20iT - 61T^{2}
67 1+7.37T+67T2 1 + 7.37T + 67T^{2}
71 1+9.16T+71T2 1 + 9.16T + 71T^{2}
73 1−9.37T+73T2 1 - 9.37T + 73T^{2}
79 1−12.7iT−79T2 1 - 12.7iT - 79T^{2}
83 1−3.16T+83T2 1 - 3.16T + 83T^{2}
89 1+6iT−89T2 1 + 6iT - 89T^{2}
97 1+4.41iT−97T2 1 + 4.41iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line