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Properties

Label 2-42-1.1-c5-0-1
Degree 22
Conductor 4242
Sign 11
Analytic cond. 6.736126.73612
Root an. cond. 2.595402.59540
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9·3-s + 16·4-s + 26·5-s − 36·6-s − 49·7-s − 64·8-s + 81·9-s − 104·10-s + 664·11-s + 144·12-s + 318·13-s + 196·14-s + 234·15-s + 256·16-s + 1.58e3·17-s − 324·18-s + 236·19-s + 416·20-s − 441·21-s − 2.65e3·22-s + 2.21e3·23-s − 576·24-s − 2.44e3·25-s − 1.27e3·26-s + 729·27-s − 784·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.465·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.328·10-s + 1.65·11-s + 0.288·12-s + 0.521·13-s + 0.267·14-s + 0.268·15-s + 1/4·16-s + 1.32·17-s − 0.235·18-s + 0.149·19-s + 0.232·20-s − 0.218·21-s − 1.16·22-s + 0.871·23-s − 0.204·24-s − 0.783·25-s − 0.369·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(42s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(42s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4242    =    2⋅3⋅72 \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 6.736126.73612
Root analytic conductor: 2.595402.59540
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 42, ( :5/2), 1)(2,\ 42,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.6041484311.604148431
L(12)L(\frac12) ≈\approx 1.6041484311.604148431
L(72)L(\frac{7}{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)
bad2 1+p2T 1 + p^{2} T
3 1−p2T 1 - p^{2} T
7 1+p2T 1 + p^{2} T
good5 1−26T+p5T2 1 - 26 T + p^{5} T^{2}
11 1−664T+p5T2 1 - 664 T + p^{5} T^{2}
13 1−318T+p5T2 1 - 318 T + p^{5} T^{2}
17 1−1582T+p5T2 1 - 1582 T + p^{5} T^{2}
19 1−236T+p5T2 1 - 236 T + p^{5} T^{2}
23 1−2212T+p5T2 1 - 2212 T + p^{5} T^{2}
29 1+4954T+p5T2 1 + 4954 T + p^{5} T^{2}
31 1+7128T+p5T2 1 + 7128 T + p^{5} T^{2}
37 1−4358T+p5T2 1 - 4358 T + p^{5} T^{2}
41 1−10542T+p5T2 1 - 10542 T + p^{5} T^{2}
43 1+8452T+p5T2 1 + 8452 T + p^{5} T^{2}
47 1−5352T+p5T2 1 - 5352 T + p^{5} T^{2}
53 1+33354T+p5T2 1 + 33354 T + p^{5} T^{2}
59 1+15436T+p5T2 1 + 15436 T + p^{5} T^{2}
61 1+36762T+p5T2 1 + 36762 T + p^{5} T^{2}
67 1−40972T+p5T2 1 - 40972 T + p^{5} T^{2}
71 1+9092T+p5T2 1 + 9092 T + p^{5} T^{2}
73 1+73454T+p5T2 1 + 73454 T + p^{5} T^{2}
79 1−89400T+p5T2 1 - 89400 T + p^{5} T^{2}
83 1+6428T+p5T2 1 + 6428 T + p^{5} T^{2}
89 1+122658T+p5T2 1 + 122658 T + p^{5} T^{2}
97 1−21370T+p5T2 1 - 21370 T + p^{5} T^{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

−14.96066037683515807518789254277, −14.02558213272790024076451752766, −12.62420720661657490498070096154, −11.24976535444859459911334072796, −9.702904309423403175237025968529, −9.042382273677078234612860967986, −7.48292139659386949088251555634, −6.06450826353750466648866217386, −3.52314199137743227571797404234, −1.45318512148560827288835303215, 1.45318512148560827288835303215, 3.52314199137743227571797404234, 6.06450826353750466648866217386, 7.48292139659386949088251555634, 9.042382273677078234612860967986, 9.702904309423403175237025968529, 11.24976535444859459911334072796, 12.62420720661657490498070096154, 14.02558213272790024076451752766, 14.96066037683515807518789254277

Graph of the ZZ-function along the critical line