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Properties

Label 2-435-1.1-c3-0-28
Degree 22
Conductor 435435
Sign −1-1
Analytic cond. 25.665825.6658
Root an. cond. 5.066145.06614
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  − 4.76·2-s − 3·3-s + 14.6·4-s − 5·5-s + 14.2·6-s + 35.2·7-s − 31.8·8-s + 9·9-s + 23.8·10-s − 41.1·11-s − 44.0·12-s − 79.0·13-s − 167.·14-s + 15·15-s + 34.1·16-s + 27.4·17-s − 42.8·18-s + 142.·19-s − 73.4·20-s − 105.·21-s + 196.·22-s − 125.·23-s + 95.4·24-s + 25·25-s + 376.·26-s − 27·27-s + 517.·28-s + ⋯
L(s)  = 1  − 1.68·2-s − 0.577·3-s + 1.83·4-s − 0.447·5-s + 0.972·6-s + 1.90·7-s − 1.40·8-s + 0.333·9-s + 0.753·10-s − 1.12·11-s − 1.05·12-s − 1.68·13-s − 3.20·14-s + 0.258·15-s + 0.533·16-s + 0.391·17-s − 0.561·18-s + 1.72·19-s − 0.820·20-s − 1.09·21-s + 1.89·22-s − 1.13·23-s + 0.812·24-s + 0.200·25-s + 2.83·26-s − 0.192·27-s + 3.49·28-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 25.665825.6658
Root analytic conductor: 5.066145.06614
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 435, ( :3/2), −1)(2,\ 435,\ (\ :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
29 1−29T 1 - 29T
good2 1+4.76T+8T2 1 + 4.76T + 8T^{2}
7 1−35.2T+343T2 1 - 35.2T + 343T^{2}
11 1+41.1T+1.33e3T2 1 + 41.1T + 1.33e3T^{2}
13 1+79.0T+2.19e3T2 1 + 79.0T + 2.19e3T^{2}
17 1−27.4T+4.91e3T2 1 - 27.4T + 4.91e3T^{2}
19 1−142.T+6.85e3T2 1 - 142.T + 6.85e3T^{2}
23 1+125.T+1.21e4T2 1 + 125.T + 1.21e4T^{2}
31 1−63.9T+2.97e4T2 1 - 63.9T + 2.97e4T^{2}
37 1+39.2T+5.06e4T2 1 + 39.2T + 5.06e4T^{2}
41 1−385.T+6.89e4T2 1 - 385.T + 6.89e4T^{2}
43 1+534.T+7.95e4T2 1 + 534.T + 7.95e4T^{2}
47 1−212.T+1.03e5T2 1 - 212.T + 1.03e5T^{2}
53 1+505.T+1.48e5T2 1 + 505.T + 1.48e5T^{2}
59 1+184.T+2.05e5T2 1 + 184.T + 2.05e5T^{2}
61 1+56.9T+2.26e5T2 1 + 56.9T + 2.26e5T^{2}
67 1−313.T+3.00e5T2 1 - 313.T + 3.00e5T^{2}
71 1+163.T+3.57e5T2 1 + 163.T + 3.57e5T^{2}
73 1−695.T+3.89e5T2 1 - 695.T + 3.89e5T^{2}
79 1+1.09e3T+4.93e5T2 1 + 1.09e3T + 4.93e5T^{2}
83 1+571.T+5.71e5T2 1 + 571.T + 5.71e5T^{2}
89 1−1.38e3T+7.04e5T2 1 - 1.38e3T + 7.04e5T^{2}
97 1+573.T+9.12e5T2 1 + 573.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

−10.22194392280061065018941467545, −9.513348498867647610739672898011, −8.094729364441957749078802615091, −7.83421917774038893596698102036, −7.15928592817148718621916201143, −5.43137721193709529404816428607, −4.69255868965361302978212719829, −2.49340377944610569412060072768, −1.29947420673431791649719015475, 0, 1.29947420673431791649719015475, 2.49340377944610569412060072768, 4.69255868965361302978212719829, 5.43137721193709529404816428607, 7.15928592817148718621916201143, 7.83421917774038893596698102036, 8.094729364441957749078802615091, 9.513348498867647610739672898011, 10.22194392280061065018941467545

Graph of the ZZ-function along the critical line