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Properties

Label 2-735-1.1-c3-0-75
Degree 22
Conductor 735735
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.53·2-s − 3·3-s + 12.5·4-s − 5·5-s − 13.5·6-s + 20.5·8-s + 9·9-s − 22.6·10-s − 19.0·11-s − 37.5·12-s + 2.93·13-s + 15·15-s − 7.21·16-s + 6.49·17-s + 40.7·18-s + 5.43·19-s − 62.6·20-s − 86.3·22-s + 49.3·23-s − 61.5·24-s + 25·25-s + 13.3·26-s − 27·27-s − 291.·29-s + 67.9·30-s − 244.·31-s − 196.·32-s + ⋯
L(s)  = 1  + 1.60·2-s − 0.577·3-s + 1.56·4-s − 0.447·5-s − 0.924·6-s + 0.907·8-s + 0.333·9-s − 0.716·10-s − 0.522·11-s − 0.904·12-s + 0.0626·13-s + 0.258·15-s − 0.112·16-s + 0.0927·17-s + 0.533·18-s + 0.0656·19-s − 0.700·20-s − 0.837·22-s + 0.447·23-s − 0.523·24-s + 0.200·25-s + 0.100·26-s − 0.192·27-s − 1.86·29-s + 0.413·30-s − 1.41·31-s − 1.08·32-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: −1-1
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: 11
Selberg data: (2, 735, ( :3/2), −1)(2,\ 735,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1−4.53T+8T2 1 - 4.53T + 8T^{2}
11 1+19.0T+1.33e3T2 1 + 19.0T + 1.33e3T^{2}
13 1−2.93T+2.19e3T2 1 - 2.93T + 2.19e3T^{2}
17 1−6.49T+4.91e3T2 1 - 6.49T + 4.91e3T^{2}
19 1−5.43T+6.85e3T2 1 - 5.43T + 6.85e3T^{2}
23 1−49.3T+1.21e4T2 1 - 49.3T + 1.21e4T^{2}
29 1+291.T+2.43e4T2 1 + 291.T + 2.43e4T^{2}
31 1+244.T+2.97e4T2 1 + 244.T + 2.97e4T^{2}
37 1+193.T+5.06e4T2 1 + 193.T + 5.06e4T^{2}
41 1+315.T+6.89e4T2 1 + 315.T + 6.89e4T^{2}
43 1+300.T+7.95e4T2 1 + 300.T + 7.95e4T^{2}
47 1+86.5T+1.03e5T2 1 + 86.5T + 1.03e5T^{2}
53 1−509.T+1.48e5T2 1 - 509.T + 1.48e5T^{2}
59 1−83.3T+2.05e5T2 1 - 83.3T + 2.05e5T^{2}
61 1−5.25T+2.26e5T2 1 - 5.25T + 2.26e5T^{2}
67 1−205.T+3.00e5T2 1 - 205.T + 3.00e5T^{2}
71 1−1.00e3T+3.57e5T2 1 - 1.00e3T + 3.57e5T^{2}
73 1−1.00e3T+3.89e5T2 1 - 1.00e3T + 3.89e5T^{2}
79 1+863.T+4.93e5T2 1 + 863.T + 4.93e5T^{2}
83 1+1.33e3T+5.71e5T2 1 + 1.33e3T + 5.71e5T^{2}
89 1+326.T+7.04e5T2 1 + 326.T + 7.04e5T^{2}
97 1+1.52e3T+9.12e5T2 1 + 1.52e3T + 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

−9.788231791734121133244800726488, −8.583567694137355609637747489301, −7.34849276792725376954747804927, −6.75053180079583229876213778141, −5.52897637370693726493914237096, −5.19352521014216754559126703206, −4.02174794598969722967063116783, −3.30289378211385297672103041551, −1.91758650570189710748989195870, 0, 1.91758650570189710748989195870, 3.30289378211385297672103041551, 4.02174794598969722967063116783, 5.19352521014216754559126703206, 5.52897637370693726493914237096, 6.75053180079583229876213778141, 7.34849276792725376954747804927, 8.583567694137355609637747489301, 9.788231791734121133244800726488

Graph of the ZZ-function along the critical line