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Properties

Label 2-1152-8.5-c3-0-28
Degree 22
Conductor 11521152
Sign 11
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  + 17.8i·5-s − 22.6·7-s − 44.2i·11-s + 17.8i·13-s − 70·17-s − 82.2i·19-s + 158.·23-s − 195.·25-s + 125. i·29-s − 404. i·35-s − 375. i·37-s + 182·41-s − 132. i·43-s − 316.·47-s + 169.·49-s + ⋯
L(s)  = 1  + 1.59i·5-s − 1.22·7-s − 1.21i·11-s + 0.381i·13-s − 0.998·17-s − 0.992i·19-s + 1.43·23-s − 1.56·25-s + 0.801i·29-s − 1.95i·35-s − 1.66i·37-s + 0.693·41-s − 0.471i·43-s − 0.983·47-s + 0.492·49-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 11
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), 1)(2,\ 1152,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3141600531.314160053
L(12)L(\frac12) ≈\approx 1.3141600531.314160053
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
good5 1−17.8iT−125T2 1 - 17.8iT - 125T^{2}
7 1+22.6T+343T2 1 + 22.6T + 343T^{2}
11 1+44.2iT−1.33e3T2 1 + 44.2iT - 1.33e3T^{2}
13 1−17.8iT−2.19e3T2 1 - 17.8iT - 2.19e3T^{2}
17 1+70T+4.91e3T2 1 + 70T + 4.91e3T^{2}
19 1+82.2iT−6.85e3T2 1 + 82.2iT - 6.85e3T^{2}
23 1−158.T+1.21e4T2 1 - 158.T + 1.21e4T^{2}
29 1−125.iT−2.43e4T2 1 - 125. iT - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1+375.iT−5.06e4T2 1 + 375. iT - 5.06e4T^{2}
41 1−182T+6.89e4T2 1 - 182T + 6.89e4T^{2}
43 1+132.iT−7.95e4T2 1 + 132. iT - 7.95e4T^{2}
47 1+316.T+1.03e5T2 1 + 316.T + 1.03e5T^{2}
53 1+125.iT−1.48e5T2 1 + 125. iT - 1.48e5T^{2}
59 1−82.2iT−2.05e5T2 1 - 82.2iT - 2.05e5T^{2}
61 1−232.iT−2.26e5T2 1 - 232. iT - 2.26e5T^{2}
67 1−221.iT−3.00e5T2 1 - 221. iT - 3.00e5T^{2}
71 1−113.T+3.57e5T2 1 - 113.T + 3.57e5T^{2}
73 1−910T+3.89e5T2 1 - 910T + 3.89e5T^{2}
79 1−678.T+4.93e5T2 1 - 678.T + 4.93e5T^{2}
83 1−714.iT−5.71e5T2 1 - 714. iT - 5.71e5T^{2}
89 1−546T+7.04e5T2 1 - 546T + 7.04e5T^{2}
97 1+490T+9.12e5T2 1 + 490T + 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

−9.319191630723803869816689832846, −8.867088036165347399582991566143, −7.50994398730473879851690874466, −6.63726675884539034109106469708, −6.50934994326694564621360151637, −5.31008946437703544524167275836, −3.83455927725786060119868363253, −3.10430814164095505884132287060, −2.42838453537155661987102857042, −0.47231757526067068572610863332, 0.70515643916818248835163208312, 1.88337866719055649217695386256, 3.22418600486866050122847410967, 4.40899517226769976302220527345, 4.96347475837193427802990757268, 6.05830774221144873085812928053, 6.86214296114207818759319386300, 7.921430874119922447323193223720, 8.673684342845594809792989673379, 9.636510526807099895753218902705

Graph of the ZZ-function along the critical line