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Properties

Label 2-1152-12.11-c1-0-3
Degree 22
Conductor 11521152
Sign 0.577−0.816i0.577 - 0.816i
Analytic cond. 9.198769.19876
Root an. cond. 3.032943.03294
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  − 3.41i·5-s + 4.82i·7-s − 2.82·11-s + 2.82·13-s + 5.41i·17-s + 5.65i·19-s + 1.17·23-s − 6.65·25-s − 0.585i·29-s − 3.17i·31-s + 16.4·35-s + 3.65·37-s + 2.58i·41-s + 9.65i·43-s + 12.4·47-s + ⋯
L(s)  = 1  − 1.52i·5-s + 1.82i·7-s − 0.852·11-s + 0.784·13-s + 1.31i·17-s + 1.29i·19-s + 0.244·23-s − 1.33·25-s − 0.108i·29-s − 0.569i·31-s + 2.78·35-s + 0.601·37-s + 0.403i·41-s + 1.47i·43-s + 1.82·47-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((0.577−0.816i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+1/2) L(s)=((0.577−0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 0.577−0.816i0.577 - 0.816i
Analytic conductor: 9.198769.19876
Root analytic conductor: 3.032943.03294
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1152(1151,⋅)\chi_{1152} (1151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :1/2), 0.577−0.816i)(2,\ 1152,\ (\ :1/2),\ 0.577 - 0.816i)

Particular Values

L(1)L(1) ≈\approx 1.3945437761.394543776
L(12)L(\frac12) ≈\approx 1.3945437761.394543776
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
3 1 1
good5 1+3.41iT−5T2 1 + 3.41iT - 5T^{2}
7 1−4.82iT−7T2 1 - 4.82iT - 7T^{2}
11 1+2.82T+11T2 1 + 2.82T + 11T^{2}
13 1−2.82T+13T2 1 - 2.82T + 13T^{2}
17 1−5.41iT−17T2 1 - 5.41iT - 17T^{2}
19 1−5.65iT−19T2 1 - 5.65iT - 19T^{2}
23 1−1.17T+23T2 1 - 1.17T + 23T^{2}
29 1+0.585iT−29T2 1 + 0.585iT - 29T^{2}
31 1+3.17iT−31T2 1 + 3.17iT - 31T^{2}
37 1−3.65T+37T2 1 - 3.65T + 37T^{2}
41 1−2.58iT−41T2 1 - 2.58iT - 41T^{2}
43 1−9.65iT−43T2 1 - 9.65iT - 43T^{2}
47 1−12.4T+47T2 1 - 12.4T + 47T^{2}
53 1−5.07iT−53T2 1 - 5.07iT - 53T^{2}
59 1−2.34T+59T2 1 - 2.34T + 59T^{2}
61 1−7.65T+61T2 1 - 7.65T + 61T^{2}
67 1+12iT−67T2 1 + 12iT - 67T^{2}
71 1+4.48T+71T2 1 + 4.48T + 71T^{2}
73 1−4T+73T2 1 - 4T + 73T^{2}
79 1−6.48iT−79T2 1 - 6.48iT - 79T^{2}
83 1+5.17T+83T2 1 + 5.17T + 83T^{2}
89 1−12.2iT−89T2 1 - 12.2iT - 89T^{2}
97 1−13.6T+97T2 1 - 13.6T + 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

−9.659782263365283782899747319839, −9.006824808807019217940217507063, −8.265305353930769001933405575848, −7.989776243437302024940850642402, −6.07729898028013508789540023631, −5.77471564526972110438194069593, −4.92732105823723651188326058489, −3.87139355916767467356030409684, −2.46970747226928753047365743098, −1.38847680784358031150558796350, 0.64734792517724004595993496822, 2.53141991830990794282502796096, 3.36945640649224550823612231383, 4.29028695802397280129765324100, 5.42205513459670754041838068018, 6.76024066656569152880103174150, 7.07957111567331679685021720726, 7.67198198379906614342749987224, 8.890880769226326988577320052968, 10.01331724837133053638583730926

Graph of the ZZ-function along the critical line