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Properties

Label 2-650-1.1-c5-0-45
Degree 22
Conductor 650650
Sign 11
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
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 + 19.2·3-s + 16·4-s − 76.8·6-s + 51.1·7-s − 64·8-s + 126.·9-s + 494.·11-s + 307.·12-s + 169·13-s − 204.·14-s + 256·16-s + 2.36e3·17-s − 506.·18-s − 699.·19-s + 983.·21-s − 1.97e3·22-s + 3.80e3·23-s − 1.23e3·24-s − 676·26-s − 2.23e3·27-s + 818.·28-s + 2.26e3·29-s − 9.08e3·31-s − 1.02e3·32-s + 9.50e3·33-s − 9.45e3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.23·3-s + 0.5·4-s − 0.872·6-s + 0.394·7-s − 0.353·8-s + 0.520·9-s + 1.23·11-s + 0.616·12-s + 0.277·13-s − 0.279·14-s + 0.250·16-s + 1.98·17-s − 0.368·18-s − 0.444·19-s + 0.486·21-s − 0.871·22-s + 1.50·23-s − 0.436·24-s − 0.196·26-s − 0.590·27-s + 0.197·28-s + 0.499·29-s − 1.69·31-s − 0.176·32-s + 1.51·33-s − 1.40·34-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 650, ( :5/2), 1)(2,\ 650,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.2395945693.239594569
L(12)L(\frac12) ≈\approx 3.2395945693.239594569
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 1
13 1−169T 1 - 169T
good3 1−19.2T+243T2 1 - 19.2T + 243T^{2}
7 1−51.1T+1.68e4T2 1 - 51.1T + 1.68e4T^{2}
11 1−494.T+1.61e5T2 1 - 494.T + 1.61e5T^{2}
17 1−2.36e3T+1.41e6T2 1 - 2.36e3T + 1.41e6T^{2}
19 1+699.T+2.47e6T2 1 + 699.T + 2.47e6T^{2}
23 1−3.80e3T+6.43e6T2 1 - 3.80e3T + 6.43e6T^{2}
29 1−2.26e3T+2.05e7T2 1 - 2.26e3T + 2.05e7T^{2}
31 1+9.08e3T+2.86e7T2 1 + 9.08e3T + 2.86e7T^{2}
37 1−4.48e3T+6.93e7T2 1 - 4.48e3T + 6.93e7T^{2}
41 1+1.24e4T+1.15e8T2 1 + 1.24e4T + 1.15e8T^{2}
43 1−1.25e4T+1.47e8T2 1 - 1.25e4T + 1.47e8T^{2}
47 1+3.60e3T+2.29e8T2 1 + 3.60e3T + 2.29e8T^{2}
53 1+7.54e3T+4.18e8T2 1 + 7.54e3T + 4.18e8T^{2}
59 1−5.00e4T+7.14e8T2 1 - 5.00e4T + 7.14e8T^{2}
61 1+2.47e4T+8.44e8T2 1 + 2.47e4T + 8.44e8T^{2}
67 1+4.37e4T+1.35e9T2 1 + 4.37e4T + 1.35e9T^{2}
71 1−7.60e3T+1.80e9T2 1 - 7.60e3T + 1.80e9T^{2}
73 1−2.16e4T+2.07e9T2 1 - 2.16e4T + 2.07e9T^{2}
79 1−1.03e5T+3.07e9T2 1 - 1.03e5T + 3.07e9T^{2}
83 1−8.03e4T+3.93e9T2 1 - 8.03e4T + 3.93e9T^{2}
89 1−5.93e4T+5.58e9T2 1 - 5.93e4T + 5.58e9T^{2}
97 1−7.34e4T+8.58e9T2 1 - 7.34e4T + 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

−9.386052843136875448954230395613, −9.031040630806040890852339833700, −8.125042653218812375459479426261, −7.50101419147195880763578107508, −6.49123300356450636589429254814, −5.28156226622721029818777836363, −3.78155342309024774301903675927, −3.09395232492462549755281601555, −1.82559770030333400897496479536, −0.967778186990377222386137072651, 0.967778186990377222386137072651, 1.82559770030333400897496479536, 3.09395232492462549755281601555, 3.78155342309024774301903675927, 5.28156226622721029818777836363, 6.49123300356450636589429254814, 7.50101419147195880763578107508, 8.125042653218812375459479426261, 9.031040630806040890852339833700, 9.386052843136875448954230395613

Graph of the ZZ-function along the critical line