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Properties

Label 2-1224-8.5-c1-0-69
Degree 22
Conductor 12241224
Sign −0.366+0.930i-0.366 + 0.930i
Analytic cond. 9.773689.77368
Root an. cond. 3.126293.12629
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.548i)2-s + (1.39 − 1.43i)4-s − 3.97i·5-s + 3.04·7-s + (−1.03 + 2.63i)8-s + (2.18 + 5.17i)10-s − 5.52i·11-s − 4.79i·13-s + (−3.97 + 1.67i)14-s + (−0.0952 − 3.99i)16-s + 17-s − 1.88i·19-s + (−5.68 − 5.55i)20-s + (3.03 + 7.19i)22-s + 2.95·23-s + ⋯
L(s)  = 1  + (−0.921 + 0.388i)2-s + (0.698 − 0.715i)4-s − 1.77i·5-s + 1.15·7-s + (−0.366 + 0.930i)8-s + (0.689 + 1.63i)10-s − 1.66i·11-s − 1.33i·13-s + (−1.06 + 0.447i)14-s + (−0.0238 − 0.999i)16-s + 0.242·17-s − 0.432i·19-s + (−1.27 − 1.24i)20-s + (0.646 + 1.53i)22-s + 0.616·23-s + ⋯

Functional equation

Λ(s)=(1224s/2 ΓC(s) L(s)=((−0.366+0.930i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1224s/2 ΓC(s+1/2) L(s)=((−0.366+0.930i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12241224    =    23⋅32⋅172^{3} \cdot 3^{2} \cdot 17
Sign: −0.366+0.930i-0.366 + 0.930i
Analytic conductor: 9.773689.77368
Root analytic conductor: 3.126293.12629
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1224(613,⋅)\chi_{1224} (613, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1224, ( :1/2), −0.366+0.930i)(2,\ 1224,\ (\ :1/2),\ -0.366 + 0.930i)

Particular Values

L(1)L(1) ≈\approx 1.1561403781.156140378
L(12)L(\frac12) ≈\approx 1.1561403781.156140378
L(32)L(\frac{3}{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+(1.30−0.548i)T 1 + (1.30 - 0.548i)T
3 1 1
17 1−T 1 - T
good5 1+3.97iT−5T2 1 + 3.97iT - 5T^{2}
7 1−3.04T+7T2 1 - 3.04T + 7T^{2}
11 1+5.52iT−11T2 1 + 5.52iT - 11T^{2}
13 1+4.79iT−13T2 1 + 4.79iT - 13T^{2}
19 1+1.88iT−19T2 1 + 1.88iT - 19T^{2}
23 1−2.95T+23T2 1 - 2.95T + 23T^{2}
29 1−8.15iT−29T2 1 - 8.15iT - 29T^{2}
31 1−0.314T+31T2 1 - 0.314T + 31T^{2}
37 1−3.47iT−37T2 1 - 3.47iT - 37T^{2}
41 1−6.67T+41T2 1 - 6.67T + 41T^{2}
43 1+0.0362iT−43T2 1 + 0.0362iT - 43T^{2}
47 1+5.44T+47T2 1 + 5.44T + 47T^{2}
53 1−6.18iT−53T2 1 - 6.18iT - 53T^{2}
59 1−2.22iT−59T2 1 - 2.22iT - 59T^{2}
61 1−8.22iT−61T2 1 - 8.22iT - 61T^{2}
67 1−9.92iT−67T2 1 - 9.92iT - 67T^{2}
71 1−5.84T+71T2 1 - 5.84T + 71T^{2}
73 1−5.30T+73T2 1 - 5.30T + 73T^{2}
79 1+11.3T+79T2 1 + 11.3T + 79T^{2}
83 1+14.0iT−83T2 1 + 14.0iT - 83T^{2}
89 1+10.5T+89T2 1 + 10.5T + 89T^{2}
97 1−3.72T+97T2 1 - 3.72T + 97T^{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

−9.066051799977121236531678365095, −8.577176640175170277643047891288, −8.186600400808549149388372397765, −7.37903256952464666458332967967, −5.88261450700795440487623652418, −5.39638598791969017917790347291, −4.67611463438695438985768206449, −3.04242623065013232909199565434, −1.37272046307045913689727966719, −0.72895845405003357304716028319, 1.85489704554639559712522504767, 2.33878817392372017523651888654, 3.68185882355856584619918696184, 4.61240051478577739191875874310, 6.22668762862457734089779926690, 6.97910581587020650951450854358, 7.50455251333977178442441594741, 8.204784087099369998107319129219, 9.544776820968218536922896081922, 9.845015037308118471190491083774

Graph of the ZZ-function along the critical line