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Properties

Label 2-435-1.1-c3-0-39
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  + 4.39·2-s + 3·3-s + 11.2·4-s − 5·5-s + 13.1·6-s + 31.5·7-s + 14.3·8-s + 9·9-s − 21.9·10-s − 5.28·11-s + 33.8·12-s − 5.98·13-s + 138.·14-s − 15·15-s − 27.0·16-s + 84.6·17-s + 39.5·18-s + 57.3·19-s − 56.3·20-s + 94.6·21-s − 23.2·22-s + 31.0·23-s + 43.1·24-s + 25·25-s − 26.2·26-s + 27·27-s + 355.·28-s + ⋯
L(s)  = 1  + 1.55·2-s + 0.577·3-s + 1.40·4-s − 0.447·5-s + 0.896·6-s + 1.70·7-s + 0.636·8-s + 0.333·9-s − 0.694·10-s − 0.144·11-s + 0.813·12-s − 0.127·13-s + 2.64·14-s − 0.258·15-s − 0.422·16-s + 1.20·17-s + 0.517·18-s + 0.692·19-s − 0.630·20-s + 0.983·21-s − 0.225·22-s + 0.281·23-s + 0.367·24-s + 0.200·25-s − 0.198·26-s + 0.192·27-s + 2.40·28-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 6.1709055976.170905597
L(12)L(\frac12) ≈\approx 6.1709055976.170905597
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 1−4.39T+8T2 1 - 4.39T + 8T^{2}
7 1−31.5T+343T2 1 - 31.5T + 343T^{2}
11 1+5.28T+1.33e3T2 1 + 5.28T + 1.33e3T^{2}
13 1+5.98T+2.19e3T2 1 + 5.98T + 2.19e3T^{2}
17 1−84.6T+4.91e3T2 1 - 84.6T + 4.91e3T^{2}
19 1−57.3T+6.85e3T2 1 - 57.3T + 6.85e3T^{2}
23 1−31.0T+1.21e4T2 1 - 31.0T + 1.21e4T^{2}
31 1+14.6T+2.97e4T2 1 + 14.6T + 2.97e4T^{2}
37 1+7.76T+5.06e4T2 1 + 7.76T + 5.06e4T^{2}
41 1+399.T+6.89e4T2 1 + 399.T + 6.89e4T^{2}
43 1+17.9T+7.95e4T2 1 + 17.9T + 7.95e4T^{2}
47 1+262.T+1.03e5T2 1 + 262.T + 1.03e5T^{2}
53 1+64.9T+1.48e5T2 1 + 64.9T + 1.48e5T^{2}
59 1−122.T+2.05e5T2 1 - 122.T + 2.05e5T^{2}
61 1−264.T+2.26e5T2 1 - 264.T + 2.26e5T^{2}
67 1−622.T+3.00e5T2 1 - 622.T + 3.00e5T^{2}
71 1−327.T+3.57e5T2 1 - 327.T + 3.57e5T^{2}
73 1+833.T+3.89e5T2 1 + 833.T + 3.89e5T^{2}
79 1+666.T+4.93e5T2 1 + 666.T + 4.93e5T^{2}
83 1+1.32e3T+5.71e5T2 1 + 1.32e3T + 5.71e5T^{2}
89 1+189.T+7.04e5T2 1 + 189.T + 7.04e5T^{2}
97 1+137.T+9.12e5T2 1 + 137.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

−11.20779340759601399211642415999, −10.01085587433024632325871053068, −8.610778346901755367248132164311, −7.84230926907683717380523110891, −6.98366238561003080368965341153, −5.47592683839105187865401654376, −4.88801250286545525653114308774, −3.89758150034432843763641614958, −2.88660655055716560795087740868, −1.52627884677992010900054700058, 1.52627884677992010900054700058, 2.88660655055716560795087740868, 3.89758150034432843763641614958, 4.88801250286545525653114308774, 5.47592683839105187865401654376, 6.98366238561003080368965341153, 7.84230926907683717380523110891, 8.610778346901755367248132164311, 10.01085587433024632325871053068, 11.20779340759601399211642415999

Graph of the ZZ-function along the critical line