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Properties

Label 2-3724-133.37-c0-0-0
Degree 22
Conductor 37243724
Sign −0.701−0.712i-0.701 - 0.712i
Analytic cond. 1.858511.85851
Root an. cond. 1.363271.36327
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + (−0.5 + 0.866i)61-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)81-s − 2·83-s + 0.999·85-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1 + 1.73i)23-s − 43-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + (−0.5 + 0.866i)61-s + (−0.5 − 0.866i)73-s + (−0.499 − 0.866i)81-s − 2·83-s + 0.999·85-s + ⋯

Functional equation

Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3724s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37243724    =    22⋅72⋅192^{2} \cdot 7^{2} \cdot 19
Sign: −0.701−0.712i-0.701 - 0.712i
Analytic conductor: 1.858511.85851
Root analytic conductor: 1.363271.36327
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3724(569,⋅)\chi_{3724} (569, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3724, ( :0), −0.701−0.712i)(2,\ 3724,\ (\ :0),\ -0.701 - 0.712i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79092982580.7909298258
L(12)L(\frac12) ≈\approx 0.79092982580.7909298258
L(1)L(1) 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 1
7 1 1
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1−T2 1 - T^{2}
17 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+2T+T2 1 + 2T + T^{2}
89 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
97 1−T2 1 - T^{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

−9.048616211462397993847955943489, −8.057432989909275422487080002864, −7.35778004714409328051748199250, −7.04136499346802271328214882555, −6.04244948035794181222370909854, −5.14319191255763356976977400963, −4.44912547678862069342612598912, −3.42604861988246881576847567162, −2.69553600530230036484334596130, −1.69620310924035666304499455531, 0.44657338964641443807477173670, 1.66573020750314684042702900610, 3.02763530166876607375185831068, 3.87516854703353057433343640847, 4.42941452020313430966528628575, 5.52147903420193745335290785175, 6.20290676487627219566036145459, 6.76694191394180329297765370560, 8.030873348295406610039220135554, 8.484499916256035092218116157487

Graph of the ZZ-function along the critical line