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Properties

Label 2-195-1.1-c5-0-13
Degree 22
Conductor 195195
Sign 11
Analytic cond. 31.274831.2748
Root an. cond. 5.592395.59239
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  − 6.53·2-s + 9·3-s + 10.6·4-s − 25·5-s − 58.8·6-s + 229.·7-s + 139.·8-s + 81·9-s + 163.·10-s + 284.·11-s + 96.2·12-s − 169·13-s − 1.49e3·14-s − 225·15-s − 1.25e3·16-s + 1.58e3·17-s − 529.·18-s − 2.17e3·19-s − 267.·20-s + 2.06e3·21-s − 1.85e3·22-s + 1.12e3·23-s + 1.25e3·24-s + 625·25-s + 1.10e3·26-s + 729·27-s + 2.44e3·28-s + ⋯
L(s)  = 1  − 1.15·2-s + 0.577·3-s + 0.334·4-s − 0.447·5-s − 0.666·6-s + 1.76·7-s + 0.769·8-s + 0.333·9-s + 0.516·10-s + 0.708·11-s + 0.192·12-s − 0.277·13-s − 2.04·14-s − 0.258·15-s − 1.22·16-s + 1.32·17-s − 0.385·18-s − 1.38·19-s − 0.149·20-s + 1.01·21-s − 0.818·22-s + 0.444·23-s + 0.444·24-s + 0.200·25-s + 0.320·26-s + 0.192·27-s + 0.590·28-s + ⋯

Functional equation

Λ(s)=(195s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(195s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 195195    =    3⋅5⋅133 \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 31.274831.2748
Root analytic conductor: 5.592395.59239
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 195, ( :5/2), 1)(2,\ 195,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.5221922291.522192229
L(12)L(\frac12) ≈\approx 1.5221922291.522192229
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)
bad3 1−9T 1 - 9T
5 1+25T 1 + 25T
13 1+169T 1 + 169T
good2 1+6.53T+32T2 1 + 6.53T + 32T^{2}
7 1−229.T+1.68e4T2 1 - 229.T + 1.68e4T^{2}
11 1−284.T+1.61e5T2 1 - 284.T + 1.61e5T^{2}
17 1−1.58e3T+1.41e6T2 1 - 1.58e3T + 1.41e6T^{2}
19 1+2.17e3T+2.47e6T2 1 + 2.17e3T + 2.47e6T^{2}
23 1−1.12e3T+6.43e6T2 1 - 1.12e3T + 6.43e6T^{2}
29 1−1.54e3T+2.05e7T2 1 - 1.54e3T + 2.05e7T^{2}
31 1+3.76e3T+2.86e7T2 1 + 3.76e3T + 2.86e7T^{2}
37 1−7.40e3T+6.93e7T2 1 - 7.40e3T + 6.93e7T^{2}
41 1+1.04e4T+1.15e8T2 1 + 1.04e4T + 1.15e8T^{2}
43 1+4.09e3T+1.47e8T2 1 + 4.09e3T + 1.47e8T^{2}
47 1+4.37e3T+2.29e8T2 1 + 4.37e3T + 2.29e8T^{2}
53 1−2.70e4T+4.18e8T2 1 - 2.70e4T + 4.18e8T^{2}
59 1−2.90e4T+7.14e8T2 1 - 2.90e4T + 7.14e8T^{2}
61 1+4.18e4T+8.44e8T2 1 + 4.18e4T + 8.44e8T^{2}
67 1−3.28e4T+1.35e9T2 1 - 3.28e4T + 1.35e9T^{2}
71 1−6.94e4T+1.80e9T2 1 - 6.94e4T + 1.80e9T^{2}
73 1−6.07e4T+2.07e9T2 1 - 6.07e4T + 2.07e9T^{2}
79 1−4.43e4T+3.07e9T2 1 - 4.43e4T + 3.07e9T^{2}
83 1−4.76e3T+3.93e9T2 1 - 4.76e3T + 3.93e9T^{2}
89 1−4.57e4T+5.58e9T2 1 - 4.57e4T + 5.58e9T^{2}
97 1+1.66e5T+8.58e9T2 1 + 1.66e5T + 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

−11.32952578594256432972320943534, −10.53985567168061452565547128906, −9.437630493872892147972184468424, −8.398799374891618187844901138770, −8.039365078772438658627480418585, −7.01848904783579682512590348488, −5.03254307490986481327045238858, −3.98892755298893769311121505235, −2.00060033565779870236323301538, −0.957867069670270083209848156613, 0.957867069670270083209848156613, 2.00060033565779870236323301538, 3.98892755298893769311121505235, 5.03254307490986481327045238858, 7.01848904783579682512590348488, 8.039365078772438658627480418585, 8.398799374891618187844901138770, 9.437630493872892147972184468424, 10.53985567168061452565547128906, 11.32952578594256432972320943534

Graph of the ZZ-function along the critical line