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Properties

Label 2-1872-52.23-c0-0-2
Degree 22
Conductor 18721872
Sign 0.859+0.511i0.859 + 0.511i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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)7-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s + 25-s + 31-s + (−1.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s − 1.73i·79-s + (0.499 + 0.866i)91-s + (1.5 + 0.866i)97-s + 1.73i·103-s + 1.73i·109-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)13-s + (1 − 1.73i)19-s + 25-s + 31-s + (−1.5 − 0.866i)43-s + (−0.5 + 0.866i)61-s + (0.5 + 0.866i)67-s − 1.73i·73-s − 1.73i·79-s + (0.499 + 0.866i)91-s + (1.5 + 0.866i)97-s + 1.73i·103-s + 1.73i·109-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.859+0.511i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=((0.859+0.511i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.859+0.511i0.859 + 0.511i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1872(127,⋅)\chi_{1872} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :0), 0.859+0.511i)(2,\ 1872,\ (\ :0),\ 0.859 + 0.511i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2157429331.215742933
L(12)L(\frac12) ≈\approx 1.2157429331.215742933
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
3 1 1
13 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1−T2 1 - T^{2}
7 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)T2 1 + (-0.5 + 0.866i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
19 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1−T+T2 1 - T + T^{2}
37 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
41 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
43 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + 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)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
73 1+1.73iT−T2 1 + 1.73iT - T^{2}
79 1+1.73iT−T2 1 + 1.73iT - T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−1.5−0.866i)T+(0.5+0.866i)T2 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)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

−9.270122457354307106114620230900, −8.688493767395100143807968470441, −7.59900747279923966940697472165, −7.09541984055325404482419459072, −6.36324519490532612749923817031, −4.95094137534503040634238591449, −4.68298980380325634009599432509, −3.48949418039190592007887775730, −2.41945539463594212418224734452, −1.06138918427718956768636872321, 1.42447038187771156563102882191, 2.64956560146176588859850882204, 3.50350428789934877603123737609, 4.80047423870362881622086045835, 5.41385126673182048669050671591, 6.18205017944184376491011712210, 7.21578883377869743361612671285, 8.151379180294265031923847796239, 8.438880104983498679039953818229, 9.653846259463153295125125448199

Graph of the ZZ-function along the critical line