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Properties

Label 2-735-1.1-c3-0-34
Degree 22
Conductor 735735
Sign 11
Analytic cond. 43.366443.3664
Root an. cond. 6.585316.58531
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.80·2-s + 3·3-s + 15.1·4-s + 5·5-s − 14.4·6-s − 34.1·8-s + 9·9-s − 24.0·10-s + 53.2·11-s + 45.3·12-s + 60.9·13-s + 15·15-s + 43.2·16-s + 31.1·17-s − 43.2·18-s − 17.9·19-s + 75.5·20-s − 255.·22-s − 34.8·23-s − 102.·24-s + 25·25-s − 292.·26-s + 27·27-s + 141.·29-s − 72.0·30-s + 117.·31-s + 65.1·32-s + ⋯
L(s)  = 1  − 1.69·2-s + 0.577·3-s + 1.88·4-s + 0.447·5-s − 0.981·6-s − 1.50·8-s + 0.333·9-s − 0.759·10-s + 1.45·11-s + 1.08·12-s + 1.30·13-s + 0.258·15-s + 0.675·16-s + 0.443·17-s − 0.566·18-s − 0.216·19-s + 0.844·20-s − 2.48·22-s − 0.315·23-s − 0.871·24-s + 0.200·25-s − 2.20·26-s + 0.192·27-s + 0.905·29-s − 0.438·30-s + 0.682·31-s + 0.360·32-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 43.366443.3664
Root analytic conductor: 6.585316.58531
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 735, ( :3/2), 1)(2,\ 735,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5369504431.536950443
L(12)L(\frac12) ≈\approx 1.5369504431.536950443
L(52)L(\frac{5}{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 1−3T 1 - 3T
5 1−5T 1 - 5T
7 1 1
good2 1+4.80T+8T2 1 + 4.80T + 8T^{2}
11 1−53.2T+1.33e3T2 1 - 53.2T + 1.33e3T^{2}
13 1−60.9T+2.19e3T2 1 - 60.9T + 2.19e3T^{2}
17 1−31.1T+4.91e3T2 1 - 31.1T + 4.91e3T^{2}
19 1+17.9T+6.85e3T2 1 + 17.9T + 6.85e3T^{2}
23 1+34.8T+1.21e4T2 1 + 34.8T + 1.21e4T^{2}
29 1−141.T+2.43e4T2 1 - 141.T + 2.43e4T^{2}
31 1−117.T+2.97e4T2 1 - 117.T + 2.97e4T^{2}
37 1−175.T+5.06e4T2 1 - 175.T + 5.06e4T^{2}
41 1−411.T+6.89e4T2 1 - 411.T + 6.89e4T^{2}
43 1+498.T+7.95e4T2 1 + 498.T + 7.95e4T^{2}
47 1+290.T+1.03e5T2 1 + 290.T + 1.03e5T^{2}
53 1−582.T+1.48e5T2 1 - 582.T + 1.48e5T^{2}
59 1+657.T+2.05e5T2 1 + 657.T + 2.05e5T^{2}
61 1+417.T+2.26e5T2 1 + 417.T + 2.26e5T^{2}
67 1+567.T+3.00e5T2 1 + 567.T + 3.00e5T^{2}
71 1−887.T+3.57e5T2 1 - 887.T + 3.57e5T^{2}
73 1+1.09e3T+3.89e5T2 1 + 1.09e3T + 3.89e5T^{2}
79 1−135.T+4.93e5T2 1 - 135.T + 4.93e5T^{2}
83 1+464.T+5.71e5T2 1 + 464.T + 5.71e5T^{2}
89 1−31.8T+7.04e5T2 1 - 31.8T + 7.04e5T^{2}
97 1+254.T+9.12e5T2 1 + 254.T + 9.12e5T^{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

−9.780433935760052100696124456133, −9.049578756549598300935307921827, −8.529018963271609250773449045931, −7.73440268675887245164458385735, −6.63557797302969230360466167108, −6.12141400302357032994485880994, −4.27530223112602512493758069265, −3.02662306448184205829162452995, −1.70016164869208188515200772240, −0.981736770778698454641450064248, 0.981736770778698454641450064248, 1.70016164869208188515200772240, 3.02662306448184205829162452995, 4.27530223112602512493758069265, 6.12141400302357032994485880994, 6.63557797302969230360466167108, 7.73440268675887245164458385735, 8.529018963271609250773449045931, 9.049578756549598300935307921827, 9.780433935760052100696124456133

Graph of the ZZ-function along the critical line