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Properties

Label 2-1224-1224.515-c0-0-0
Degree 22
Conductor 12241224
Sign 0.206−0.978i0.206 - 0.978i
Analytic cond. 0.6108550.610855
Root an. cond. 0.7815720.781572
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  + (−0.130 + 0.991i)2-s + (0.991 + 0.130i)3-s + (−0.965 − 0.258i)4-s + (−0.258 + 0.965i)6-s + (0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.0578 − 0.117i)11-s + (−0.923 − 0.382i)12-s + (0.866 + 0.5i)16-s + (0.608 − 0.793i)17-s + (−0.382 + 0.923i)18-s + (0.739 + 1.78i)19-s + (0.123 − 0.0420i)22-s + (0.5 − 0.866i)24-s + (−0.793 + 0.608i)25-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (0.991 + 0.130i)3-s + (−0.965 − 0.258i)4-s + (−0.258 + 0.965i)6-s + (0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.0578 − 0.117i)11-s + (−0.923 − 0.382i)12-s + (0.866 + 0.5i)16-s + (0.608 − 0.793i)17-s + (−0.382 + 0.923i)18-s + (0.739 + 1.78i)19-s + (0.123 − 0.0420i)22-s + (0.5 − 0.866i)24-s + (−0.793 + 0.608i)25-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((0.206−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s) L(s)=((0.206−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: 0.206−0.978i0.206 - 0.978i
Analytic conductor: 0.6108550.610855
Root analytic conductor: 0.7815720.781572
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1224(515,⋅)\chi_{1224} (515, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :0), 0.206−0.978i)(2,\ 1224,\ (\ :0),\ 0.206 - 0.978i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2955128321.295512832
L(12)L(\frac12) ≈\approx 1.2955128321.295512832
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+(0.130−0.991i)T 1 + (0.130 - 0.991i)T
3 1+(−0.991−0.130i)T 1 + (-0.991 - 0.130i)T
17 1+(−0.608+0.793i)T 1 + (-0.608 + 0.793i)T
good5 1+(0.793−0.608i)T2 1 + (0.793 - 0.608i)T^{2}
7 1+(−0.793−0.608i)T2 1 + (-0.793 - 0.608i)T^{2}
11 1+(0.0578+0.117i)T+(−0.608+0.793i)T2 1 + (0.0578 + 0.117i)T + (-0.608 + 0.793i)T^{2}
13 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
19 1+(−0.739−1.78i)T+(−0.707+0.707i)T2 1 + (-0.739 - 1.78i)T + (-0.707 + 0.707i)T^{2}
23 1+(0.991+0.130i)T2 1 + (0.991 + 0.130i)T^{2}
29 1+(−0.130−0.991i)T2 1 + (-0.130 - 0.991i)T^{2}
31 1+(−0.608−0.793i)T2 1 + (-0.608 - 0.793i)T^{2}
37 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
41 1+(1.31+1.50i)T+(−0.130+0.991i)T2 1 + (1.31 + 1.50i)T + (-0.130 + 0.991i)T^{2}
43 1+(0.741+0.965i)T+(−0.258+0.965i)T2 1 + (0.741 + 0.965i)T + (-0.258 + 0.965i)T^{2}
47 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
53 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
59 1+(−0.258+0.0340i)T+(0.965−0.258i)T2 1 + (-0.258 + 0.0340i)T + (0.965 - 0.258i)T^{2}
61 1+(0.793+0.608i)T2 1 + (0.793 + 0.608i)T^{2}
67 1+(1.37−0.793i)T+(0.5−0.866i)T2 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2}
71 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
73 1+(−0.172+0.867i)T+(−0.923−0.382i)T2 1 + (-0.172 + 0.867i)T + (-0.923 - 0.382i)T^{2}
79 1+(−0.608+0.793i)T2 1 + (-0.608 + 0.793i)T^{2}
83 1+(0.758+0.0999i)T+(0.965+0.258i)T2 1 + (0.758 + 0.0999i)T + (0.965 + 0.258i)T^{2}
89 1+(1.30+1.30i)T+iT2 1 + (1.30 + 1.30i)T + iT^{2}
97 1+(−0.991+1.13i)T+(−0.130−0.991i)T2 1 + (-0.991 + 1.13i)T + (-0.130 - 0.991i)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

−9.968628203046819118382864804446, −9.080998506726375958431379823565, −8.421568370156410314557594419969, −7.57336200572014890763822780591, −7.18173994267915470688768803744, −5.88904535071486304451085906428, −5.16719320117643873570215025064, −3.98193916461049155017443002720, −3.27731426600377709206052513393, −1.60964664061964359731485009065, 1.36133624318732821079596927104, 2.53200576275795315873008682711, 3.31858626429759564719699486054, 4.28070034350693345658547639712, 5.17465671014218782949909761886, 6.58202237038833111172480724610, 7.64327991789084615392508435128, 8.286659135915719940776013280192, 9.046542854076860664164623611213, 9.785008326618562371883244296943

Graph of the ZZ-function along the critical line