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Properties

Label 2-3267-11.8-c0-0-3
Degree 22
Conductor 32673267
Sign −0.969−0.245i-0.969 - 0.245i
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  + (0.309 − 0.951i)4-s + (−1.83 − 0.596i)7-s + (0.304 + 0.418i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (−1.13 + 1.56i)28-s + (−1.40 + 1.01i)31-s − 1.41i·43-s + (2.21 + 1.60i)49-s + (0.492 − 0.159i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (−0.492 − 0.159i)73-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (−1.83 − 0.596i)7-s + (0.304 + 0.418i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (−1.13 + 1.56i)28-s + (−1.40 + 1.01i)31-s − 1.41i·43-s + (2.21 + 1.60i)49-s + (0.492 − 0.159i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (−0.492 − 0.159i)73-s + ⋯

Functional equation

Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.969−0.245i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3267s/2 ΓC(s) L(s)=((−0.969−0.245i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32673267    =    33⋅1123^{3} \cdot 11^{2}
Sign: −0.969−0.245i-0.969 - 0.245i
Analytic conductor: 1.630441.63044
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3267(2296,⋅)\chi_{3267} (2296, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3267, ( :0), −0.969−0.245i)(2,\ 3267,\ (\ :0),\ -0.969 - 0.245i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.28140353390.2814035339
L(12)L(\frac12) ≈\approx 0.28140353390.2814035339
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 1
11 1 1
good2 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
5 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
7 1+(1.83+0.596i)T+(0.809+0.587i)T2 1 + (1.83 + 0.596i)T + (0.809 + 0.587i)T^{2}
13 1+(−0.304−0.418i)T+(−0.309+0.951i)T2 1 + (-0.304 - 0.418i)T + (-0.309 + 0.951i)T^{2}
17 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
19 1+(1.34−0.437i)T+(0.809−0.587i)T2 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2}
23 1+T2 1 + T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
31 1+(1.40−1.01i)T+(0.309−0.951i)T2 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2}
37 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
41 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
43 1+1.41iT−T2 1 + 1.41iT - T^{2}
47 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
53 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
59 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
61 1+(0.831−1.14i)T+(−0.309−0.951i)T2 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2}
67 1+1.73T+T2 1 + 1.73T + T^{2}
71 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
73 1+(0.492+0.159i)T+(0.809+0.587i)T2 1 + (0.492 + 0.159i)T + (0.809 + 0.587i)T^{2}
79 1+(0.304+0.418i)T+(−0.309+0.951i)T2 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2}
83 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
89 1+T2 1 + T^{2}
97 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)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

−8.709741581406830751127542904935, −7.36890440183870921641968434093, −6.81208981858364561726121642537, −6.19530651296143474492933288812, −5.68926862580309376658556635020, −4.43987910172009155018175336989, −3.72729047365320498732631439077, −2.73158617323826797637217215702, −1.65383512614875707759921920346, −0.15258689065323235842523057926, 2.09112482333349634024479048526, 2.98738465729378153963531781143, 3.52747787912402605060295333950, 4.39645062846893636768718656376, 5.74198513557267096752641050781, 6.25632512553464191221870340413, 6.97182912173584667892716283589, 7.67782804496646100838645389234, 8.576521189996604884775192467180, 9.179963402800902264464744338325

Graph of the ZZ-function along the critical line