Properties

Label 2-1152-8.5-c3-0-41
Degree 22
Conductor 11521152
Sign ii
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
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  − 17.4i·5-s − 2.99·7-s + 10.6i·11-s + 43.3i·13-s + 37.8·17-s − 79.8i·19-s + 191.·23-s − 178.·25-s + 138. i·29-s + 212.·31-s + 52.1i·35-s − 270. i·37-s + 441.·41-s + 64.1i·43-s + 436.·47-s + ⋯
L(s)  = 1  − 1.55i·5-s − 0.161·7-s + 0.291i·11-s + 0.924i·13-s + 0.540·17-s − 0.964i·19-s + 1.73·23-s − 1.43·25-s + 0.889i·29-s + 1.22·31-s + 0.251i·35-s − 1.20i·37-s + 1.68·41-s + 0.227i·43-s + 1.35·47-s + ⋯

Functional equation

Λ(s)=(1152s/2ΓC(s)L(s)=(iΛ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2ΓC(s+3/2)L(s)=(iΛ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27322^{7} \cdot 3^{2}
Sign: ii
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1152(577,)\chi_{1152} (577, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), i)(2,\ 1152,\ (\ :3/2),\ i)

Particular Values

L(2)L(2) \approx 2.0245183032.024518303
L(12)L(\frac12) \approx 2.0245183032.024518303
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
good5 1+17.4iT125T2 1 + 17.4iT - 125T^{2}
7 1+2.99T+343T2 1 + 2.99T + 343T^{2}
11 110.6iT1.33e3T2 1 - 10.6iT - 1.33e3T^{2}
13 143.3iT2.19e3T2 1 - 43.3iT - 2.19e3T^{2}
17 137.8T+4.91e3T2 1 - 37.8T + 4.91e3T^{2}
19 1+79.8iT6.85e3T2 1 + 79.8iT - 6.85e3T^{2}
23 1191.T+1.21e4T2 1 - 191.T + 1.21e4T^{2}
29 1138.iT2.43e4T2 1 - 138. iT - 2.43e4T^{2}
31 1212.T+2.97e4T2 1 - 212.T + 2.97e4T^{2}
37 1+270.iT5.06e4T2 1 + 270. iT - 5.06e4T^{2}
41 1441.T+6.89e4T2 1 - 441.T + 6.89e4T^{2}
43 164.1iT7.95e4T2 1 - 64.1iT - 7.95e4T^{2}
47 1436.T+1.03e5T2 1 - 436.T + 1.03e5T^{2}
53 1+278.iT1.48e5T2 1 + 278. iT - 1.48e5T^{2}
59 1+830.iT2.05e5T2 1 + 830. iT - 2.05e5T^{2}
61 1724.iT2.26e5T2 1 - 724. iT - 2.26e5T^{2}
67 1+859.iT3.00e5T2 1 + 859. iT - 3.00e5T^{2}
71 1+681.T+3.57e5T2 1 + 681.T + 3.57e5T^{2}
73 1+785.T+3.89e5T2 1 + 785.T + 3.89e5T^{2}
79 1+1.01e3T+4.93e5T2 1 + 1.01e3T + 4.93e5T^{2}
83 1467.iT5.71e5T2 1 - 467. iT - 5.71e5T^{2}
89 1+510.T+7.04e5T2 1 + 510.T + 7.04e5T^{2}
97 1+234.T+9.12e5T2 1 + 234.T + 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.082317791964432618961436192461, −8.696897889485989905644804789777, −7.58020778845096596192826576124, −6.78865748945506162008002634882, −5.64391547531652107107325396367, −4.82315385099333061410254280144, −4.24454605072518016685984935119, −2.85243219494961954132908540998, −1.49508914497466477179276019444, −0.59200752991177752465189296636, 1.03831991580905578803413510145, 2.74695332104695814375032805201, 3.09723682831516419102028626658, 4.28852507159995542693777041533, 5.66678552318488415548793401747, 6.23351540323450038393346052382, 7.21511579718168299776845782335, 7.78261672797151904285233618058, 8.768573805642631131465774585551, 9.965612169736428387958457661649

Graph of the ZZ-function along the critical line