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Properties

Label 2-4805-1.1-c1-0-171
Degree 22
Conductor 48054805
Sign 11
Analytic cond. 38.368138.3681
Root an. cond. 6.194206.19420
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.737·2-s + 2.97·3-s − 1.45·4-s + 5-s + 2.19·6-s + 1.93·7-s − 2.54·8-s + 5.84·9-s + 0.737·10-s − 2.66·11-s − 4.32·12-s − 2.07·13-s + 1.42·14-s + 2.97·15-s + 1.03·16-s + 7.48·17-s + 4.30·18-s − 2.49·19-s − 1.45·20-s + 5.74·21-s − 1.96·22-s + 7.64·23-s − 7.58·24-s + 25-s − 1.52·26-s + 8.45·27-s − 2.81·28-s + ⋯
L(s)  = 1  + 0.521·2-s + 1.71·3-s − 0.727·4-s + 0.447·5-s + 0.895·6-s + 0.729·7-s − 0.901·8-s + 1.94·9-s + 0.233·10-s − 0.802·11-s − 1.24·12-s − 0.575·13-s + 0.380·14-s + 0.767·15-s + 0.257·16-s + 1.81·17-s + 1.01·18-s − 0.573·19-s − 0.325·20-s + 1.25·21-s − 0.418·22-s + 1.59·23-s − 1.54·24-s + 0.200·25-s − 0.300·26-s + 1.62·27-s − 0.531·28-s + ⋯

Functional equation

Λ(s)=(4805s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4805s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48054805    =    5⋅3125 \cdot 31^{2}
Sign: 11
Analytic conductor: 38.368138.3681
Root analytic conductor: 6.194206.19420
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4805, ( :1/2), 1)(2,\ 4805,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.7357196044.735719604
L(12)L(\frac12) ≈\approx 4.7357196044.735719604
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)
bad5 1−T 1 - T
31 1 1
good2 1−0.737T+2T2 1 - 0.737T + 2T^{2}
3 1−2.97T+3T2 1 - 2.97T + 3T^{2}
7 1−1.93T+7T2 1 - 1.93T + 7T^{2}
11 1+2.66T+11T2 1 + 2.66T + 11T^{2}
13 1+2.07T+13T2 1 + 2.07T + 13T^{2}
17 1−7.48T+17T2 1 - 7.48T + 17T^{2}
19 1+2.49T+19T2 1 + 2.49T + 19T^{2}
23 1−7.64T+23T2 1 - 7.64T + 23T^{2}
29 1+3.59T+29T2 1 + 3.59T + 29T^{2}
37 1+2.84T+37T2 1 + 2.84T + 37T^{2}
41 1−9.86T+41T2 1 - 9.86T + 41T^{2}
43 1−1.32T+43T2 1 - 1.32T + 43T^{2}
47 1+1.50T+47T2 1 + 1.50T + 47T^{2}
53 1−9.18T+53T2 1 - 9.18T + 53T^{2}
59 1−12.3T+59T2 1 - 12.3T + 59T^{2}
61 1−2.67T+61T2 1 - 2.67T + 61T^{2}
67 1+0.787T+67T2 1 + 0.787T + 67T^{2}
71 1+5.23T+71T2 1 + 5.23T + 71T^{2}
73 1+12.4T+73T2 1 + 12.4T + 73T^{2}
79 1+10.5T+79T2 1 + 10.5T + 79T^{2}
83 1−8.30T+83T2 1 - 8.30T + 83T^{2}
89 1−6.26T+89T2 1 - 6.26T + 89T^{2}
97 1−10.1T+97T2 1 - 10.1T + 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

−8.433860657723231893392582126626, −7.62292483146976901265889889678, −7.22728580724597919154212077221, −5.81500537580298051795654953383, −5.18064586656044320927353665485, −4.51034333335736371037543913532, −3.62276092161798429277131379976, −2.95638102978990703490022486352, −2.25981729075605843069802413780, −1.09737330259781234847808189479, 1.09737330259781234847808189479, 2.25981729075605843069802413780, 2.95638102978990703490022486352, 3.62276092161798429277131379976, 4.51034333335736371037543913532, 5.18064586656044320927353665485, 5.81500537580298051795654953383, 7.22728580724597919154212077221, 7.62292483146976901265889889678, 8.433860657723231893392582126626

Graph of the ZZ-function along the critical line