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Properties

Label 2-735-1.1-c3-0-77
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  + 3.44·2-s + 3·3-s + 3.89·4-s − 5·5-s + 10.3·6-s − 14.1·8-s + 9·9-s − 17.2·10-s − 5.84·11-s + 11.6·12-s − 3.12·13-s − 15·15-s − 79.9·16-s − 62.1·17-s + 31.0·18-s + 17.6·19-s − 19.4·20-s − 20.1·22-s − 112.·23-s − 42.4·24-s + 25·25-s − 10.7·26-s + 27·27-s − 164.·29-s − 51.7·30-s + 20.2·31-s − 162.·32-s + ⋯
L(s)  = 1  + 1.21·2-s + 0.577·3-s + 0.486·4-s − 0.447·5-s + 0.703·6-s − 0.625·8-s + 0.333·9-s − 0.545·10-s − 0.160·11-s + 0.280·12-s − 0.0667·13-s − 0.258·15-s − 1.24·16-s − 0.886·17-s + 0.406·18-s + 0.212·19-s − 0.217·20-s − 0.195·22-s − 1.01·23-s − 0.361·24-s + 0.200·25-s − 0.0814·26-s + 0.192·27-s − 1.05·29-s − 0.314·30-s + 0.117·31-s − 0.898·32-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 1−3.44T+8T2 1 - 3.44T + 8T^{2}
11 1+5.84T+1.33e3T2 1 + 5.84T + 1.33e3T^{2}
13 1+3.12T+2.19e3T2 1 + 3.12T + 2.19e3T^{2}
17 1+62.1T+4.91e3T2 1 + 62.1T + 4.91e3T^{2}
19 1−17.6T+6.85e3T2 1 - 17.6T + 6.85e3T^{2}
23 1+112.T+1.21e4T2 1 + 112.T + 1.21e4T^{2}
29 1+164.T+2.43e4T2 1 + 164.T + 2.43e4T^{2}
31 1−20.2T+2.97e4T2 1 - 20.2T + 2.97e4T^{2}
37 1+300.T+5.06e4T2 1 + 300.T + 5.06e4T^{2}
41 1−83.8T+6.89e4T2 1 - 83.8T + 6.89e4T^{2}
43 1+44.5T+7.95e4T2 1 + 44.5T + 7.95e4T^{2}
47 1−88.6T+1.03e5T2 1 - 88.6T + 1.03e5T^{2}
53 1+363.T+1.48e5T2 1 + 363.T + 1.48e5T^{2}
59 1+660.T+2.05e5T2 1 + 660.T + 2.05e5T^{2}
61 1−805.T+2.26e5T2 1 - 805.T + 2.26e5T^{2}
67 1+510.T+3.00e5T2 1 + 510.T + 3.00e5T^{2}
71 1+615.T+3.57e5T2 1 + 615.T + 3.57e5T^{2}
73 1−30.6T+3.89e5T2 1 - 30.6T + 3.89e5T^{2}
79 1−235.T+4.93e5T2 1 - 235.T + 4.93e5T^{2}
83 1+229.T+5.71e5T2 1 + 229.T + 5.71e5T^{2}
89 1−1.46e3T+7.04e5T2 1 - 1.46e3T + 7.04e5T^{2}
97 1+490.T+9.12e5T2 1 + 490.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

−9.433452754860636125538637852185, −8.706367676524001665999933296999, −7.74457059182801060959936831783, −6.79133305509100288127586120452, −5.81035291381217053551635350786, −4.78731193275618642709179724220, −3.99015148140149828481422327872, −3.17506575025664656211138791098, −2.03418902156462835054619168296, 0, 2.03418902156462835054619168296, 3.17506575025664656211138791098, 3.99015148140149828481422327872, 4.78731193275618642709179724220, 5.81035291381217053551635350786, 6.79133305509100288127586120452, 7.74457059182801060959936831783, 8.706367676524001665999933296999, 9.433452754860636125538637852185

Graph of the ZZ-function along the critical line