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Properties

Label 2-435-1.1-c3-0-14
Degree 22
Conductor 435435
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.40·2-s − 3·3-s − 6.01·4-s + 5·5-s − 4.22·6-s + 22.4·7-s − 19.7·8-s + 9·9-s + 7.04·10-s + 11.9·11-s + 18.0·12-s − 24.3·13-s + 31.5·14-s − 15·15-s + 20.2·16-s − 57.0·17-s + 12.6·18-s − 101.·19-s − 30.0·20-s − 67.2·21-s + 16.8·22-s + 133.·23-s + 59.2·24-s + 25·25-s − 34.3·26-s − 27·27-s − 134.·28-s + ⋯
L(s)  = 1  + 0.498·2-s − 0.577·3-s − 0.751·4-s + 0.447·5-s − 0.287·6-s + 1.21·7-s − 0.872·8-s + 0.333·9-s + 0.222·10-s + 0.327·11-s + 0.434·12-s − 0.520·13-s + 0.602·14-s − 0.258·15-s + 0.317·16-s − 0.814·17-s + 0.166·18-s − 1.22·19-s − 0.336·20-s − 0.698·21-s + 0.163·22-s + 1.20·23-s + 0.503·24-s + 0.200·25-s − 0.259·26-s − 0.192·27-s − 0.909·28-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: 11
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: 00
Selberg data: (2, 435, ( :3/2), 1)(2,\ 435,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.9185508501.918550850
L(12)L(\frac12) ≈\approx 1.9185508501.918550850
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 1−1.40T+8T2 1 - 1.40T + 8T^{2}
7 1−22.4T+343T2 1 - 22.4T + 343T^{2}
11 1−11.9T+1.33e3T2 1 - 11.9T + 1.33e3T^{2}
13 1+24.3T+2.19e3T2 1 + 24.3T + 2.19e3T^{2}
17 1+57.0T+4.91e3T2 1 + 57.0T + 4.91e3T^{2}
19 1+101.T+6.85e3T2 1 + 101.T + 6.85e3T^{2}
23 1−133.T+1.21e4T2 1 - 133.T + 1.21e4T^{2}
31 1−292.T+2.97e4T2 1 - 292.T + 2.97e4T^{2}
37 1−393.T+5.06e4T2 1 - 393.T + 5.06e4T^{2}
41 1−237.T+6.89e4T2 1 - 237.T + 6.89e4T^{2}
43 1+82.3T+7.95e4T2 1 + 82.3T + 7.95e4T^{2}
47 1−490.T+1.03e5T2 1 - 490.T + 1.03e5T^{2}
53 1+416.T+1.48e5T2 1 + 416.T + 1.48e5T^{2}
59 1−320.T+2.05e5T2 1 - 320.T + 2.05e5T^{2}
61 1−612.T+2.26e5T2 1 - 612.T + 2.26e5T^{2}
67 1−569.T+3.00e5T2 1 - 569.T + 3.00e5T^{2}
71 1+689.T+3.57e5T2 1 + 689.T + 3.57e5T^{2}
73 1−125.T+3.89e5T2 1 - 125.T + 3.89e5T^{2}
79 1+356.T+4.93e5T2 1 + 356.T + 4.93e5T^{2}
83 1−947.T+5.71e5T2 1 - 947.T + 5.71e5T^{2}
89 1+1.21e3T+7.04e5T2 1 + 1.21e3T + 7.04e5T^{2}
97 1+597.T+9.12e5T2 1 + 597.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

−10.90105425819941109377575027626, −9.849260834025950715096617286944, −8.909804316935680525189515357544, −8.095981467397186707971108894432, −6.74243392330184941112889953837, −5.79396820707187029202082573448, −4.74269334162501638398106232719, −4.32976011783828265162166917631, −2.47698213947610411168312872873, −0.880889658107417387830054797693, 0.880889658107417387830054797693, 2.47698213947610411168312872873, 4.32976011783828265162166917631, 4.74269334162501638398106232719, 5.79396820707187029202082573448, 6.74243392330184941112889953837, 8.095981467397186707971108894432, 8.909804316935680525189515357544, 9.849260834025950715096617286944, 10.90105425819941109377575027626

Graph of the ZZ-function along the critical line