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Properties

Label 2-55-1.1-c3-0-9
Degree 22
Conductor 5555
Sign −1-1
Analytic cond. 3.245103.24510
Root an. cond. 1.801411.80141
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s − 7·4-s − 5·5-s − 3·6-s − 9·7-s − 15·8-s − 18·9-s − 5·10-s + 11·11-s + 21·12-s + 2·13-s − 9·14-s + 15·15-s + 41·16-s + 21·17-s − 18·18-s − 85·19-s + 35·20-s + 27·21-s + 11·22-s + 22·23-s + 45·24-s + 25·25-s + 2·26-s + 135·27-s + 63·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s − 0.204·6-s − 0.485·7-s − 0.662·8-s − 2/3·9-s − 0.158·10-s + 0.301·11-s + 0.505·12-s + 0.0426·13-s − 0.171·14-s + 0.258·15-s + 0.640·16-s + 0.299·17-s − 0.235·18-s − 1.02·19-s + 0.391·20-s + 0.280·21-s + 0.106·22-s + 0.199·23-s + 0.382·24-s + 1/5·25-s + 0.0150·26-s + 0.962·27-s + 0.425·28-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: −1-1
Analytic conductor: 3.245103.24510
Root analytic conductor: 1.801411.80141
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 55, ( :3/2), −1)(2,\ 55,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad5 1+pT 1 + p T
11 1−pT 1 - p T
good2 1−T+p3T2 1 - T + p^{3} T^{2}
3 1+pT+p3T2 1 + p T + p^{3} T^{2}
7 1+9T+p3T2 1 + 9 T + p^{3} T^{2}
13 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1−21T+p3T2 1 - 21 T + p^{3} T^{2}
19 1+85T+p3T2 1 + 85 T + p^{3} T^{2}
23 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
29 1+165T+p3T2 1 + 165 T + p^{3} T^{2}
31 1+83T+p3T2 1 + 83 T + p^{3} T^{2}
37 1−T+p3T2 1 - T + p^{3} T^{2}
41 1+478T+p3T2 1 + 478 T + p^{3} T^{2}
43 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
47 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
53 1+683T+p3T2 1 + 683 T + p^{3} T^{2}
59 1+290T+p3T2 1 + 290 T + p^{3} T^{2}
61 1−257T+p3T2 1 - 257 T + p^{3} T^{2}
67 1−776T+p3T2 1 - 776 T + p^{3} T^{2}
71 1+313T+p3T2 1 + 313 T + p^{3} T^{2}
73 1−902T+p3T2 1 - 902 T + p^{3} T^{2}
79 1−830T+p3T2 1 - 830 T + p^{3} T^{2}
83 1−842T+p3T2 1 - 842 T + p^{3} T^{2}
89 1−25T+p3T2 1 - 25 T + p^{3} T^{2}
97 1+1784T+p3T2 1 + 1784 T + p^{3} 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

−14.23818884762054503300538549627, −13.01744979915157662584740759019, −12.11542215936679308087649861172, −10.91863500584103319510480633270, −9.440842169835386193433429604798, −8.273305932618668249050131927195, −6.40740806035650295877386950202, −5.10483814859930842390914923141, −3.57122137945214328716009661090, 0, 3.57122137945214328716009661090, 5.10483814859930842390914923141, 6.40740806035650295877386950202, 8.273305932618668249050131927195, 9.440842169835386193433429604798, 10.91863500584103319510480633270, 12.11542215936679308087649861172, 13.01744979915157662584740759019, 14.23818884762054503300538549627

Graph of the ZZ-function along the critical line