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Properties

Label 2-2952-41.40-c1-0-39
Degree 22
Conductor 29522952
Sign 0.0582+0.998i0.0582 + 0.998i
Analytic cond. 23.571823.5718
Root an. cond. 4.855084.85508
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  + 1.60·5-s + 0.181i·7-s − 2.05i·11-s + 0.949i·13-s − 3.83i·17-s − 6.15i·19-s + 4.73·23-s − 2.42·25-s − 3.26i·29-s − 7.12·31-s + 0.291i·35-s − 4.42·37-s + (0.372 + 6.39i)41-s − 7.54·43-s − 10.2i·47-s + ⋯
L(s)  = 1  + 0.717·5-s + 0.0687i·7-s − 0.618i·11-s + 0.263i·13-s − 0.930i·17-s − 1.41i·19-s + 0.987·23-s − 0.484·25-s − 0.605i·29-s − 1.28·31-s + 0.0493i·35-s − 0.727·37-s + (0.0582 + 0.998i)41-s − 1.15·43-s − 1.49i·47-s + ⋯

Functional equation

Λ(s)=(2952s/2 ΓC(s) L(s)=((0.0582+0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2952s/2 ΓC(s+1/2) L(s)=((0.0582+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29522952    =    23⋅32⋅412^{3} \cdot 3^{2} \cdot 41
Sign: 0.0582+0.998i0.0582 + 0.998i
Analytic conductor: 23.571823.5718
Root analytic conductor: 4.855084.85508
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2952(2377,⋅)\chi_{2952} (2377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2952, ( :1/2), 0.0582+0.998i)(2,\ 2952,\ (\ :1/2),\ 0.0582 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.6898305021.689830502
L(12)L(\frac12) ≈\approx 1.6898305021.689830502
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
41 1+(−0.372−6.39i)T 1 + (-0.372 - 6.39i)T
good5 1−1.60T+5T2 1 - 1.60T + 5T^{2}
7 1−0.181iT−7T2 1 - 0.181iT - 7T^{2}
11 1+2.05iT−11T2 1 + 2.05iT - 11T^{2}
13 1−0.949iT−13T2 1 - 0.949iT - 13T^{2}
17 1+3.83iT−17T2 1 + 3.83iT - 17T^{2}
19 1+6.15iT−19T2 1 + 6.15iT - 19T^{2}
23 1−4.73T+23T2 1 - 4.73T + 23T^{2}
29 1+3.26iT−29T2 1 + 3.26iT - 29T^{2}
31 1+7.12T+31T2 1 + 7.12T + 31T^{2}
37 1+4.42T+37T2 1 + 4.42T + 37T^{2}
43 1+7.54T+43T2 1 + 7.54T + 43T^{2}
47 1+10.2iT−47T2 1 + 10.2iT - 47T^{2}
53 1+0.554iT−53T2 1 + 0.554iT - 53T^{2}
59 1−11.6T+59T2 1 - 11.6T + 59T^{2}
61 1+9.80T+61T2 1 + 9.80T + 61T^{2}
67 1+7.91iT−67T2 1 + 7.91iT - 67T^{2}
71 1+3.73iT−71T2 1 + 3.73iT - 71T^{2}
73 1−12.1T+73T2 1 - 12.1T + 73T^{2}
79 1+8.84iT−79T2 1 + 8.84iT - 79T^{2}
83 1−1.70T+83T2 1 - 1.70T + 83T^{2}
89 1+4.76iT−89T2 1 + 4.76iT - 89T^{2}
97 1−9.39iT−97T2 1 - 9.39iT - 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

−8.850463946504120442320849357879, −7.76628801802387251352102866586, −6.95953422222405964965002127268, −6.36977612389958824043747592158, −5.35658736002669903451218926146, −4.92911506209472018646943843468, −3.69990777061748047402783835077, −2.79364610578420736463928388320, −1.89260644021547095338657760799, −0.52066281085968050218160781695, 1.40268589954349397974121017253, 2.14981895577315241833913710910, 3.39042608811034907437630557455, 4.14137429206870761199712459216, 5.30763948230355957160423278126, 5.73829288538697141505940267409, 6.67613592817155349022040418563, 7.38821520147804814871590982405, 8.207073894096243505073139921294, 8.968720457608307240040229669649

Graph of the ZZ-function along the critical line