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Properties

Label 2-3267-27.20-c0-0-0
Degree 22
Conductor 32673267
Sign 0.993−0.116i0.993 - 0.116i
Analytic cond. 1.630441.63044
Root an. cond. 1.276881.27688
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  + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (−1.26 + 0.223i)5-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)12-s + (−1.26 − 0.223i)15-s + (0.173 − 0.984i)16-s + (−0.826 + 0.984i)20-s + (1.11 + 1.32i)23-s + (0.613 − 0.223i)25-s + (0.500 + 0.866i)27-s + (1.43 − 1.20i)31-s + 36-s + (0.766 + 1.32i)37-s + (−1.11 − 0.642i)45-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + (−1.26 + 0.223i)5-s + (0.766 + 0.642i)9-s + (0.939 − 0.342i)12-s + (−1.26 − 0.223i)15-s + (0.173 − 0.984i)16-s + (−0.826 + 0.984i)20-s + (1.11 + 1.32i)23-s + (0.613 − 0.223i)25-s + (0.500 + 0.866i)27-s + (1.43 − 1.20i)31-s + 36-s + (0.766 + 1.32i)37-s + (−1.11 − 0.642i)45-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: 0.993−0.116i0.993 - 0.116i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(1937,⋅)\chi_{3267} (1937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), 0.993−0.116i)(2,\ 3267,\ (\ :0),\ 0.993 - 0.116i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7741058161.774105816
L(12)L(\frac12) ≈\approx 1.7741058161.774105816
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)
bad3 1+(−0.939−0.342i)T 1 + (-0.939 - 0.342i)T
11 1 1
good2 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
5 1+(1.26−0.223i)T+(0.939−0.342i)T2 1 + (1.26 - 0.223i)T + (0.939 - 0.342i)T^{2}
7 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
13 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
17 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(−1.11−1.32i)T+(−0.173+0.984i)T2 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2}
29 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
31 1+(−1.43+1.20i)T+(0.173−0.984i)T2 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2}
37 1+(−0.766−1.32i)T+(−0.5+0.866i)T2 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2}
41 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
43 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
47 1+(−1.26+1.50i)T+(−0.173−0.984i)T2 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2}
53 1−0.684iT−T2 1 - 0.684iT - T^{2}
59 1+(−0.673+0.118i)T+(0.939−0.342i)T2 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2}
61 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
67 1+(1.76+0.642i)T+(0.766+0.642i)T2 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2}
71 1+(1.70−0.984i)T+(0.5−0.866i)T2 1 + (1.70 - 0.984i)T + (0.5 - 0.866i)T^{2}
73 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
79 1+(0.766−0.642i)T2 1 + (0.766 - 0.642i)T^{2}
83 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
89 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
97 1+(−0.0603+0.342i)T+(−0.939−0.342i)T2 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)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

−8.763381001254030246534307556255, −8.001643757758017533400275554669, −7.40196084390809967422998622291, −6.90175270957282161081453641918, −5.83288670562103367301706938391, −4.85364857811297964366063310576, −4.06068017359096395147018676510, −3.20059540189718734433738952548, −2.52533307895191921441656512327, −1.25772394459459126218881973657, 1.19058572886936486759117684068, 2.61979814579062458905152562597, 3.04405335082360835359014651983, 4.08162600668159128520671153046, 4.54320704668175392370062628525, 6.08516411219401110299098690591, 6.91693023974103340121958485480, 7.42213160355322890678131664661, 8.036161870456436092535755129015, 8.603832042936494714814497926793

Graph of the ZZ-function along the critical line