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Properties

Label 2-435-1.1-c3-0-15
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  − 5.05·2-s + 3·3-s + 17.5·4-s − 5·5-s − 15.1·6-s + 19.3·7-s − 48.0·8-s + 9·9-s + 25.2·10-s + 6.81·11-s + 52.5·12-s + 36.9·13-s − 97.9·14-s − 15·15-s + 102.·16-s − 71.5·17-s − 45.4·18-s + 88.3·19-s − 87.5·20-s + 58.1·21-s − 34.4·22-s + 185.·23-s − 144.·24-s + 25·25-s − 186.·26-s + 27·27-s + 339.·28-s + ⋯
L(s)  = 1  − 1.78·2-s + 0.577·3-s + 2.18·4-s − 0.447·5-s − 1.03·6-s + 1.04·7-s − 2.12·8-s + 0.333·9-s + 0.798·10-s + 0.186·11-s + 1.26·12-s + 0.788·13-s − 1.86·14-s − 0.258·15-s + 1.60·16-s − 1.02·17-s − 0.595·18-s + 1.06·19-s − 0.978·20-s + 0.604·21-s − 0.333·22-s + 1.68·23-s − 1.22·24-s + 0.200·25-s − 1.40·26-s + 0.192·27-s + 2.29·28-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 1.1400321561.140032156
L(12)L(\frac12) ≈\approx 1.1400321561.140032156
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+5.05T+8T2 1 + 5.05T + 8T^{2}
7 1−19.3T+343T2 1 - 19.3T + 343T^{2}
11 1−6.81T+1.33e3T2 1 - 6.81T + 1.33e3T^{2}
13 1−36.9T+2.19e3T2 1 - 36.9T + 2.19e3T^{2}
17 1+71.5T+4.91e3T2 1 + 71.5T + 4.91e3T^{2}
19 1−88.3T+6.85e3T2 1 - 88.3T + 6.85e3T^{2}
23 1−185.T+1.21e4T2 1 - 185.T + 1.21e4T^{2}
31 1+120.T+2.97e4T2 1 + 120.T + 2.97e4T^{2}
37 1+117.T+5.06e4T2 1 + 117.T + 5.06e4T^{2}
41 1+229.T+6.89e4T2 1 + 229.T + 6.89e4T^{2}
43 1−66.8T+7.95e4T2 1 - 66.8T + 7.95e4T^{2}
47 1+42.0T+1.03e5T2 1 + 42.0T + 1.03e5T^{2}
53 1−9.42T+1.48e5T2 1 - 9.42T + 1.48e5T^{2}
59 1+232.T+2.05e5T2 1 + 232.T + 2.05e5T^{2}
61 1+546.T+2.26e5T2 1 + 546.T + 2.26e5T^{2}
67 1−953.T+3.00e5T2 1 - 953.T + 3.00e5T^{2}
71 1−429.T+3.57e5T2 1 - 429.T + 3.57e5T^{2}
73 1−554.T+3.89e5T2 1 - 554.T + 3.89e5T^{2}
79 1−965.T+4.93e5T2 1 - 965.T + 4.93e5T^{2}
83 1−625.T+5.71e5T2 1 - 625.T + 5.71e5T^{2}
89 1+271.T+7.04e5T2 1 + 271.T + 7.04e5T^{2}
97 1−1.16e3T+9.12e5T2 1 - 1.16e3T + 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

−10.80383635026862864231431039112, −9.490839594659093023199702024390, −8.849194633638707096754678496914, −8.207983135785276456917364210385, −7.42161245175559202207243362990, −6.62690730911304084259681656298, −4.94029591447359055997234220526, −3.35009682765527178709407456479, −1.94887964615063909224104847769, −0.912823117500316736515447719411, 0.912823117500316736515447719411, 1.94887964615063909224104847769, 3.35009682765527178709407456479, 4.94029591447359055997234220526, 6.62690730911304084259681656298, 7.42161245175559202207243362990, 8.207983135785276456917364210385, 8.849194633638707096754678496914, 9.490839594659093023199702024390, 10.80383635026862864231431039112

Graph of the ZZ-function along the critical line