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Properties

Label 2-140-1.1-c5-0-6
Degree 22
Conductor 140140
Sign 11
Analytic cond. 22.453722.4537
Root an. cond. 4.738534.73853
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  + 24.5·3-s + 25·5-s + 49·7-s + 359.·9-s + 90.2·11-s − 14.4·13-s + 613.·15-s + 407.·17-s + 2.28e3·19-s + 1.20e3·21-s − 505.·23-s + 625·25-s + 2.85e3·27-s − 3.16e3·29-s − 6.23e3·31-s + 2.21e3·33-s + 1.22e3·35-s + 5.38e3·37-s − 354.·39-s + 1.17e4·41-s − 5.82e3·43-s + 8.98e3·45-s + 7.34e3·47-s + 2.40e3·49-s + 9.99e3·51-s + 1.49e4·53-s + 2.25e3·55-s + ⋯
L(s)  = 1  + 1.57·3-s + 0.447·5-s + 0.377·7-s + 1.47·9-s + 0.224·11-s − 0.0236·13-s + 0.704·15-s + 0.341·17-s + 1.45·19-s + 0.595·21-s − 0.199·23-s + 0.200·25-s + 0.754·27-s − 0.698·29-s − 1.16·31-s + 0.354·33-s + 0.169·35-s + 0.647·37-s − 0.0373·39-s + 1.09·41-s − 0.480·43-s + 0.661·45-s + 0.485·47-s + 0.142·49-s + 0.538·51-s + 0.732·53-s + 0.100·55-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 22.453722.4537
Root analytic conductor: 4.738534.73853
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 140, ( :5/2), 1)(2,\ 140,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.8361201613.836120161
L(12)L(\frac12) ≈\approx 3.8361201613.836120161
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 1
5 1−25T 1 - 25T
7 1−49T 1 - 49T
good3 1−24.5T+243T2 1 - 24.5T + 243T^{2}
11 1−90.2T+1.61e5T2 1 - 90.2T + 1.61e5T^{2}
13 1+14.4T+3.71e5T2 1 + 14.4T + 3.71e5T^{2}
17 1−407.T+1.41e6T2 1 - 407.T + 1.41e6T^{2}
19 1−2.28e3T+2.47e6T2 1 - 2.28e3T + 2.47e6T^{2}
23 1+505.T+6.43e6T2 1 + 505.T + 6.43e6T^{2}
29 1+3.16e3T+2.05e7T2 1 + 3.16e3T + 2.05e7T^{2}
31 1+6.23e3T+2.86e7T2 1 + 6.23e3T + 2.86e7T^{2}
37 1−5.38e3T+6.93e7T2 1 - 5.38e3T + 6.93e7T^{2}
41 1−1.17e4T+1.15e8T2 1 - 1.17e4T + 1.15e8T^{2}
43 1+5.82e3T+1.47e8T2 1 + 5.82e3T + 1.47e8T^{2}
47 1−7.34e3T+2.29e8T2 1 - 7.34e3T + 2.29e8T^{2}
53 1−1.49e4T+4.18e8T2 1 - 1.49e4T + 4.18e8T^{2}
59 1+4.71e4T+7.14e8T2 1 + 4.71e4T + 7.14e8T^{2}
61 1+4.28e3T+8.44e8T2 1 + 4.28e3T + 8.44e8T^{2}
67 1−4.88e4T+1.35e9T2 1 - 4.88e4T + 1.35e9T^{2}
71 1−5.85e4T+1.80e9T2 1 - 5.85e4T + 1.80e9T^{2}
73 1+1.59e3T+2.07e9T2 1 + 1.59e3T + 2.07e9T^{2}
79 1+7.91e4T+3.07e9T2 1 + 7.91e4T + 3.07e9T^{2}
83 1+7.14e4T+3.93e9T2 1 + 7.14e4T + 3.93e9T^{2}
89 1+7.71e4T+5.58e9T2 1 + 7.71e4T + 5.58e9T^{2}
97 1−1.15e4T+8.58e9T2 1 - 1.15e4T + 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

−12.49108653150524367768195427712, −11.15155187515066393564973953431, −9.757465862979467932520795011601, −9.180044296371370195865357031808, −8.054727619267592753293839216260, −7.22511552429934842807447086281, −5.52203918705780333620702924555, −3.92346835388940290120785466087, −2.74191624921347002391444778744, −1.47287247417362097021632997009, 1.47287247417362097021632997009, 2.74191624921347002391444778744, 3.92346835388940290120785466087, 5.52203918705780333620702924555, 7.22511552429934842807447086281, 8.054727619267592753293839216260, 9.180044296371370195865357031808, 9.757465862979467932520795011601, 11.15155187515066393564973953431, 12.49108653150524367768195427712

Graph of the ZZ-function along the critical line