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Properties

Label 2-1305-1.1-c3-0-41
Degree 22
Conductor 13051305
Sign 11
Analytic cond. 76.997476.9974
Root an. cond. 8.774828.77482
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.33·2-s + 3.11·4-s − 5·5-s + 1.26·7-s − 16.2·8-s − 16.6·10-s + 35.2·11-s − 44.9·13-s + 4.21·14-s − 79.2·16-s + 17.8·17-s + 113.·19-s − 15.5·20-s + 117.·22-s − 147.·23-s + 25·25-s − 149.·26-s + 3.93·28-s + 29·29-s + 65.4·31-s − 133.·32-s + 59.4·34-s − 6.31·35-s + 399.·37-s + 377.·38-s + 81.4·40-s + 271.·41-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.389·4-s − 0.447·5-s + 0.0682·7-s − 0.719·8-s − 0.527·10-s + 0.966·11-s − 0.959·13-s + 0.0804·14-s − 1.23·16-s + 0.254·17-s + 1.36·19-s − 0.174·20-s + 1.13·22-s − 1.33·23-s + 0.200·25-s − 1.13·26-s + 0.0265·28-s + 0.185·29-s + 0.379·31-s − 0.739·32-s + 0.300·34-s − 0.0305·35-s + 1.77·37-s + 1.61·38-s + 0.321·40-s + 1.03·41-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 76.997476.9974
Root analytic conductor: 8.774828.77482
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1305, ( :3/2), 1)(2,\ 1305,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.1927179123.192717912
L(12)L(\frac12) ≈\approx 3.1927179123.192717912
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)
bad3 1 1
5 1+5T 1 + 5T
29 1−29T 1 - 29T
good2 1−3.33T+8T2 1 - 3.33T + 8T^{2}
7 1−1.26T+343T2 1 - 1.26T + 343T^{2}
11 1−35.2T+1.33e3T2 1 - 35.2T + 1.33e3T^{2}
13 1+44.9T+2.19e3T2 1 + 44.9T + 2.19e3T^{2}
17 1−17.8T+4.91e3T2 1 - 17.8T + 4.91e3T^{2}
19 1−113.T+6.85e3T2 1 - 113.T + 6.85e3T^{2}
23 1+147.T+1.21e4T2 1 + 147.T + 1.21e4T^{2}
31 1−65.4T+2.97e4T2 1 - 65.4T + 2.97e4T^{2}
37 1−399.T+5.06e4T2 1 - 399.T + 5.06e4T^{2}
41 1−271.T+6.89e4T2 1 - 271.T + 6.89e4T^{2}
43 1+242.T+7.95e4T2 1 + 242.T + 7.95e4T^{2}
47 1−374.T+1.03e5T2 1 - 374.T + 1.03e5T^{2}
53 1+580.T+1.48e5T2 1 + 580.T + 1.48e5T^{2}
59 1−803.T+2.05e5T2 1 - 803.T + 2.05e5T^{2}
61 1−842.T+2.26e5T2 1 - 842.T + 2.26e5T^{2}
67 1−266.T+3.00e5T2 1 - 266.T + 3.00e5T^{2}
71 1−732.T+3.57e5T2 1 - 732.T + 3.57e5T^{2}
73 1+227.T+3.89e5T2 1 + 227.T + 3.89e5T^{2}
79 1+680.T+4.93e5T2 1 + 680.T + 4.93e5T^{2}
83 1−433.T+5.71e5T2 1 - 433.T + 5.71e5T^{2}
89 1−1.55e3T+7.04e5T2 1 - 1.55e3T + 7.04e5T^{2}
97 1+1.02e3T+9.12e5T2 1 + 1.02e3T + 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

−9.495771679912587781012828299385, −8.345640182999610129872150476415, −7.52557176028013931632382343705, −6.59920746594028912312128283215, −5.78543400641913069919005725892, −4.91009078558169633593412049480, −4.14960242673788528294353839877, −3.39861351121221073535096759041, −2.35550989068813499222307949455, −0.75063319581679578147575481685, 0.75063319581679578147575481685, 2.35550989068813499222307949455, 3.39861351121221073535096759041, 4.14960242673788528294353839877, 4.91009078558169633593412049480, 5.78543400641913069919005725892, 6.59920746594028912312128283215, 7.52557176028013931632382343705, 8.345640182999610129872150476415, 9.495771679912587781012828299385

Graph of the ZZ-function along the critical line