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Properties

Label 2-70-1.1-c5-0-6
Degree 22
Conductor 7070
Sign 11
Analytic cond. 11.226811.2268
Root an. cond. 3.350653.35065
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 30.5·3-s + 16·4-s − 25·5-s + 122.·6-s − 49·7-s + 64·8-s + 688.·9-s − 100·10-s + 392.·11-s + 488.·12-s − 631.·13-s − 196·14-s − 763.·15-s + 256·16-s − 1.37e3·17-s + 2.75e3·18-s + 1.49e3·19-s − 400·20-s − 1.49e3·21-s + 1.56e3·22-s − 4.57e3·23-s + 1.95e3·24-s + 625·25-s − 2.52e3·26-s + 1.35e4·27-s − 784·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.95·3-s + 0.5·4-s − 0.447·5-s + 1.38·6-s − 0.377·7-s + 0.353·8-s + 2.83·9-s − 0.316·10-s + 0.977·11-s + 0.978·12-s − 1.03·13-s − 0.267·14-s − 0.875·15-s + 0.250·16-s − 1.15·17-s + 2.00·18-s + 0.951·19-s − 0.223·20-s − 0.740·21-s + 0.691·22-s − 1.80·23-s + 0.692·24-s + 0.200·25-s − 0.733·26-s + 3.59·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 11.226811.2268
Root analytic conductor: 3.350653.35065
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 70, ( :5/2), 1)(2,\ 70,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.4551640684.455164068
L(12)L(\frac12) ≈\approx 4.4551640684.455164068
L(72)L(\frac{7}{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 1−4T 1 - 4T
5 1+25T 1 + 25T
7 1+49T 1 + 49T
good3 1−30.5T+243T2 1 - 30.5T + 243T^{2}
11 1−392.T+1.61e5T2 1 - 392.T + 1.61e5T^{2}
13 1+631.T+3.71e5T2 1 + 631.T + 3.71e5T^{2}
17 1+1.37e3T+1.41e6T2 1 + 1.37e3T + 1.41e6T^{2}
19 1−1.49e3T+2.47e6T2 1 - 1.49e3T + 2.47e6T^{2}
23 1+4.57e3T+6.43e6T2 1 + 4.57e3T + 6.43e6T^{2}
29 1−2.70e3T+2.05e7T2 1 - 2.70e3T + 2.05e7T^{2}
31 1+6.93e3T+2.86e7T2 1 + 6.93e3T + 2.86e7T^{2}
37 1−1.47e3T+6.93e7T2 1 - 1.47e3T + 6.93e7T^{2}
41 1−1.47e3T+1.15e8T2 1 - 1.47e3T + 1.15e8T^{2}
43 1+1.07e4T+1.47e8T2 1 + 1.07e4T + 1.47e8T^{2}
47 1+6.47e3T+2.29e8T2 1 + 6.47e3T + 2.29e8T^{2}
53 1−3.26e3T+4.18e8T2 1 - 3.26e3T + 4.18e8T^{2}
59 1+2.92e4T+7.14e8T2 1 + 2.92e4T + 7.14e8T^{2}
61 1−3.64e4T+8.44e8T2 1 - 3.64e4T + 8.44e8T^{2}
67 1+828.T+1.35e9T2 1 + 828.T + 1.35e9T^{2}
71 1−2.80e4T+1.80e9T2 1 - 2.80e4T + 1.80e9T^{2}
73 1+7.61e4T+2.07e9T2 1 + 7.61e4T + 2.07e9T^{2}
79 1−1.07e4T+3.07e9T2 1 - 1.07e4T + 3.07e9T^{2}
83 1−9.40e4T+3.93e9T2 1 - 9.40e4T + 3.93e9T^{2}
89 1+4.35e4T+5.58e9T2 1 + 4.35e4T + 5.58e9T^{2}
97 1−3.41e4T+8.58e9T2 1 - 3.41e4T + 8.58e9T^{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

−13.94889412934630709668902917304, −12.92192468627101300965225775728, −11.88806597068515224093009565664, −10.02108248265011488039728458141, −9.057862781152959972362994168154, −7.80795271579086658016837742813, −6.81345856429996430442023192835, −4.38709060751967225066505483538, −3.39080156827967761729995215346, −2.04716824787560649346165555337, 2.04716824787560649346165555337, 3.39080156827967761729995215346, 4.38709060751967225066505483538, 6.81345856429996430442023192835, 7.80795271579086658016837742813, 9.057862781152959972362994168154, 10.02108248265011488039728458141, 11.88806597068515224093009565664, 12.92192468627101300965225775728, 13.94889412934630709668902917304

Graph of the ZZ-function along the critical line