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Properties

Label 2-42e2-7.4-c3-0-39
Degree 22
Conductor 17641764
Sign 0.605+0.795i0.605 + 0.795i
Analytic cond. 104.079104.079
Root an. cond. 10.201910.2019
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 5.19i)5-s + (18 − 31.1i)11-s + 62·13-s + (57 − 98.7i)17-s + (38 + 65.8i)19-s + (−12 − 20.7i)23-s + (44.5 − 77.0i)25-s − 54·29-s + (56 − 96.9i)31-s + (89 + 154. i)37-s − 378·41-s − 172·43-s + (−96 − 166. i)47-s + (−201 + 348. i)53-s + 216·55-s + ⋯
L(s)  = 1  + (0.268 + 0.464i)5-s + (0.493 − 0.854i)11-s + 1.32·13-s + (0.813 − 1.40i)17-s + (0.458 + 0.794i)19-s + (−0.108 − 0.188i)23-s + (0.355 − 0.616i)25-s − 0.345·29-s + (0.324 − 0.561i)31-s + (0.395 + 0.684i)37-s − 1.43·41-s − 0.609·43-s + (−0.297 − 0.516i)47-s + (−0.520 + 0.902i)53-s + 0.529·55-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.605+0.795i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+3/2) L(s)=((0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.605+0.795i0.605 + 0.795i
Analytic conductor: 104.079104.079
Root analytic conductor: 10.201910.2019
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1764(361,⋅)\chi_{1764} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :3/2), 0.605+0.795i)(2,\ 1764,\ (\ :3/2),\ 0.605 + 0.795i)

Particular Values

L(2)L(2) ≈\approx 2.5379842252.537984225
L(12)L(\frac12) ≈\approx 2.5379842252.537984225
L(52)L(\frac{5}{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 1
3 1 1
7 1 1
good5 1+(−3−5.19i)T+(−62.5+108.i)T2 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2}
11 1+(−18+31.1i)T+(−665.5−1.15e3i)T2 1 + (-18 + 31.1i)T + (-665.5 - 1.15e3i)T^{2}
13 1−62T+2.19e3T2 1 - 62T + 2.19e3T^{2}
17 1+(−57+98.7i)T+(−2.45e3−4.25e3i)T2 1 + (-57 + 98.7i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(−38−65.8i)T+(−3.42e3+5.94e3i)T2 1 + (-38 - 65.8i)T + (-3.42e3 + 5.94e3i)T^{2}
23 1+(12+20.7i)T+(−6.08e3+1.05e4i)T2 1 + (12 + 20.7i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+54T+2.43e4T2 1 + 54T + 2.43e4T^{2}
31 1+(−56+96.9i)T+(−1.48e4−2.57e4i)T2 1 + (-56 + 96.9i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(−89−154.i)T+(−2.53e4+4.38e4i)T2 1 + (-89 - 154. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+378T+6.89e4T2 1 + 378T + 6.89e4T^{2}
43 1+172T+7.95e4T2 1 + 172T + 7.95e4T^{2}
47 1+(96+166.i)T+(−5.19e4+8.99e4i)T2 1 + (96 + 166. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(201−348.i)T+(−7.44e4−1.28e5i)T2 1 + (201 - 348. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(−198+342.i)T+(−1.02e5−1.77e5i)T2 1 + (-198 + 342. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(127+219.i)T+(−1.13e5+1.96e5i)T2 1 + (127 + 219. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(−506+876.i)T+(−1.50e5−2.60e5i)T2 1 + (-506 + 876. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+840T+3.57e5T2 1 + 840T + 3.57e5T^{2}
73 1+(445−770.i)T+(−1.94e5−3.36e5i)T2 1 + (445 - 770. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(40+69.2i)T+(−2.46e5+4.26e5i)T2 1 + (40 + 69.2i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−108T+5.71e5T2 1 - 108T + 5.71e5T^{2}
89 1+(819+1.41e3i)T+(−3.52e5+6.10e5i)T2 1 + (819 + 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−1.01e3T+9.12e5T2 1 - 1.01e3T + 9.12e5T^{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

−8.721452343943278549383145304237, −8.152136421632625136733224169077, −7.17816029611407992122616799436, −6.30815119834359154151319725987, −5.78773001286347016533481159173, −4.74165644082363399631367881562, −3.53044786399619899466539462131, −3.02855194034514112733958211792, −1.61695183772888529540617440184, −0.59878550989175080128156856499, 1.12547451513669133892225638442, 1.75236900299777579815509179670, 3.24262779659524419082455593518, 4.02088769281063797096125660513, 5.01606429151151892555810103087, 5.82410703443406472023309587185, 6.61889124880372168107656230822, 7.47029990351427067753005347970, 8.441200011104557433124453276930, 8.949089067676813534953711212213

Graph of the ZZ-function along the critical line