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Properties

Label 2-440-1.1-c5-0-34
Degree 22
Conductor 440440
Sign 11
Analytic cond. 70.568870.5688
Root an. cond. 8.400528.40052
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  + 23·3-s + 25·5-s + 211·7-s + 286·9-s + 121·11-s + 26·13-s + 575·15-s − 407·17-s + 1.78e3·19-s + 4.85e3·21-s − 6·23-s + 625·25-s + 989·27-s − 2.38e3·29-s + 9.45e3·31-s + 2.78e3·33-s + 5.27e3·35-s − 6.91e3·37-s + 598·39-s − 9.77e3·41-s + 3.10e3·43-s + 7.15e3·45-s − 1.42e4·47-s + 2.77e4·49-s − 9.36e3·51-s − 1.86e4·53-s + 3.02e3·55-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.447·5-s + 1.62·7-s + 1.17·9-s + 0.301·11-s + 0.0426·13-s + 0.659·15-s − 0.341·17-s + 1.13·19-s + 2.40·21-s − 0.00236·23-s + 1/5·25-s + 0.261·27-s − 0.527·29-s + 1.76·31-s + 0.444·33-s + 0.727·35-s − 0.830·37-s + 0.0629·39-s − 0.908·41-s + 0.256·43-s + 0.526·45-s − 0.943·47-s + 1.64·49-s − 0.503·51-s − 0.912·53-s + 0.134·55-s + ⋯

Functional equation

Λ(s)=(440s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(440s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 440440    =    23⋅5⋅112^{3} \cdot 5 \cdot 11
Sign: 11
Analytic conductor: 70.568870.5688
Root analytic conductor: 8.400528.40052
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 440, ( :5/2), 1)(2,\ 440,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.3115372585.311537258
L(12)L(\frac12) ≈\approx 5.3115372585.311537258
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 1
5 1−p2T 1 - p^{2} T
11 1−p2T 1 - p^{2} T
good3 1−23T+p5T2 1 - 23 T + p^{5} T^{2}
7 1−211T+p5T2 1 - 211 T + p^{5} T^{2}
13 1−2pT+p5T2 1 - 2 p T + p^{5} T^{2}
17 1+407T+p5T2 1 + 407 T + p^{5} T^{2}
19 1−1789T+p5T2 1 - 1789 T + p^{5} T^{2}
23 1+6T+p5T2 1 + 6 T + p^{5} T^{2}
29 1+2387T+p5T2 1 + 2387 T + p^{5} T^{2}
31 1−9453T+p5T2 1 - 9453 T + p^{5} T^{2}
37 1+6917T+p5T2 1 + 6917 T + p^{5} T^{2}
41 1+9774T+p5T2 1 + 9774 T + p^{5} T^{2}
43 1−3108T+p5T2 1 - 3108 T + p^{5} T^{2}
47 1+14290T+p5T2 1 + 14290 T + p^{5} T^{2}
53 1+18665T+p5T2 1 + 18665 T + p^{5} T^{2}
59 1−36646T+p5T2 1 - 36646 T + p^{5} T^{2}
61 1+22945T+p5T2 1 + 22945 T + p^{5} T^{2}
67 1−35848T+p5T2 1 - 35848 T + p^{5} T^{2}
71 1−16647T+p5T2 1 - 16647 T + p^{5} T^{2}
73 1+34642T+p5T2 1 + 34642 T + p^{5} T^{2}
79 1−20554T+p5T2 1 - 20554 T + p^{5} T^{2}
83 1−7674T+p5T2 1 - 7674 T + p^{5} T^{2}
89 1+1111pT+p5T2 1 + 1111 p T + p^{5} T^{2}
97 1−55764T+p5T2 1 - 55764 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

−10.09405335402369317229011225089, −9.264345873612174735494058124536, −8.397898461653583449554085284599, −7.913183398574320457725525101521, −6.86484878389279693177263196831, −5.36703887888344359758437728356, −4.41194922639604512638898655282, −3.20771176597360810875101133196, −2.10139879047337931195073853998, −1.28486874002085073181358472049, 1.28486874002085073181358472049, 2.10139879047337931195073853998, 3.20771176597360810875101133196, 4.41194922639604512638898655282, 5.36703887888344359758437728356, 6.86484878389279693177263196831, 7.913183398574320457725525101521, 8.397898461653583449554085284599, 9.264345873612174735494058124536, 10.09405335402369317229011225089

Graph of the ZZ-function along the critical line