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Properties

Label 2-1152-8.5-c3-0-4
Degree 22
Conductor 11521152
Sign −0.707−0.707i-0.707 - 0.707i
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  + 4i·5-s − 92i·13-s − 94·17-s + 109·25-s + 284i·29-s + 396i·37-s + 230·41-s − 343·49-s − 572i·53-s + 468i·61-s + 368·65-s − 1.09e3·73-s − 376i·85-s − 1.67e3·89-s − 594·97-s + ⋯
L(s)  = 1  + 0.357i·5-s − 1.96i·13-s − 1.34·17-s + 0.871·25-s + 1.81i·29-s + 1.75i·37-s + 0.876·41-s − 49-s − 1.48i·53-s + 0.982i·61-s + 0.702·65-s − 1.76·73-s − 0.479i·85-s − 1.98·89-s − 0.621·97-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: −0.707−0.707i-0.707 - 0.707i
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), −0.707−0.707i)(2,\ 1152,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) ≈\approx 0.65663255270.6566325527
L(12)L(\frac12) ≈\approx 0.65663255270.6566325527
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 1−4iT−125T2 1 - 4iT - 125T^{2}
7 1+343T2 1 + 343T^{2}
11 1−1.33e3T2 1 - 1.33e3T^{2}
13 1+92iT−2.19e3T2 1 + 92iT - 2.19e3T^{2}
17 1+94T+4.91e3T2 1 + 94T + 4.91e3T^{2}
19 1−6.85e3T2 1 - 6.85e3T^{2}
23 1+1.21e4T2 1 + 1.21e4T^{2}
29 1−284iT−2.43e4T2 1 - 284iT - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1−396iT−5.06e4T2 1 - 396iT - 5.06e4T^{2}
41 1−230T+6.89e4T2 1 - 230T + 6.89e4T^{2}
43 1−7.95e4T2 1 - 7.95e4T^{2}
47 1+1.03e5T2 1 + 1.03e5T^{2}
53 1+572iT−1.48e5T2 1 + 572iT - 1.48e5T^{2}
59 1−2.05e5T2 1 - 2.05e5T^{2}
61 1−468iT−2.26e5T2 1 - 468iT - 2.26e5T^{2}
67 1−3.00e5T2 1 - 3.00e5T^{2}
71 1+3.57e5T2 1 + 3.57e5T^{2}
73 1+1.09e3T+3.89e5T2 1 + 1.09e3T + 3.89e5T^{2}
79 1+4.93e5T2 1 + 4.93e5T^{2}
83 1−5.71e5T2 1 - 5.71e5T^{2}
89 1+1.67e3T+7.04e5T2 1 + 1.67e3T + 7.04e5T^{2}
97 1+594T+9.12e5T2 1 + 594T + 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

−9.846721667013930985143600456161, −8.770756777207221019181577247295, −8.186543443507164219810449822741, −7.18335257154799982672761882074, −6.47652226528099738896484356665, −5.44298612168964414983687634898, −4.67019392969899697858875913045, −3.34910024260917695180001711153, −2.66915897433095959557832077903, −1.18098686291335506144820966956, 0.16134030992495487365135090624, 1.65348811902961442924815887329, 2.57686930541565764485128091400, 4.19187371631371860693197531896, 4.47175676802858234722726450670, 5.81449674838299737723670751083, 6.63330759005090478821176952031, 7.36071712365127121825596315504, 8.475707962570768734630610448643, 9.145224804552795251189529639975

Graph of the ZZ-function along the critical line