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Properties

Label 2-3840-1.1-c1-0-50
Degree 22
Conductor 38403840
Sign −1-1
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 2·7-s + 9-s − 0.828·11-s + 0.828·13-s − 15-s + 2.82·17-s − 2·21-s − 5.65·23-s + 25-s + 27-s + 3.65·29-s − 1.17·31-s − 0.828·33-s + 2·35-s − 6.48·37-s + 0.828·39-s − 3.65·41-s + 1.65·43-s − 45-s + 1.65·47-s − 3·49-s + 2.82·51-s − 11.6·53-s + 0.828·55-s + 4.82·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.755·7-s + 0.333·9-s − 0.249·11-s + 0.229·13-s − 0.258·15-s + 0.685·17-s − 0.436·21-s − 1.17·23-s + 0.200·25-s + 0.192·27-s + 0.679·29-s − 0.210·31-s − 0.144·33-s + 0.338·35-s − 1.06·37-s + 0.132·39-s − 0.571·41-s + 0.252·43-s − 0.149·45-s + 0.241·47-s − 0.428·49-s + 0.396·51-s − 1.60·53-s + 0.111·55-s + 0.628·59-s + ⋯

Functional equation

Λ(s)=(3840s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3840s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38403840    =    28⋅3⋅52^{8} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3840, ( :1/2), −1)(2,\ 3840,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
3 1−T 1 - T
5 1+T 1 + T
good7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+0.828T+11T2 1 + 0.828T + 11T^{2}
13 1−0.828T+13T2 1 - 0.828T + 13T^{2}
17 1−2.82T+17T2 1 - 2.82T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+5.65T+23T2 1 + 5.65T + 23T^{2}
29 1−3.65T+29T2 1 - 3.65T + 29T^{2}
31 1+1.17T+31T2 1 + 1.17T + 31T^{2}
37 1+6.48T+37T2 1 + 6.48T + 37T^{2}
41 1+3.65T+41T2 1 + 3.65T + 41T^{2}
43 1−1.65T+43T2 1 - 1.65T + 43T^{2}
47 1−1.65T+47T2 1 - 1.65T + 47T^{2}
53 1+11.6T+53T2 1 + 11.6T + 53T^{2}
59 1−4.82T+59T2 1 - 4.82T + 59T^{2}
61 1−9.65T+61T2 1 - 9.65T + 61T^{2}
67 1+9.65T+67T2 1 + 9.65T + 67T^{2}
71 1+13.6T+71T2 1 + 13.6T + 71T^{2}
73 1−9.31T+73T2 1 - 9.31T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+4.34T+89T2 1 + 4.34T + 89T^{2}
97 1−7.65T+97T2 1 - 7.65T + 97T^{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

−8.162244655544744862092529867049, −7.48206244061132668655557186412, −6.72299191914191127814883950879, −5.97784527345172530967753789503, −5.06421808955392790753205419238, −4.07509436663480773299586041954, −3.41947163587125376660483214850, −2.67529815770567707012049907024, −1.48670746216841464543140866380, 0, 1.48670746216841464543140866380, 2.67529815770567707012049907024, 3.41947163587125376660483214850, 4.07509436663480773299586041954, 5.06421808955392790753205419238, 5.97784527345172530967753789503, 6.72299191914191127814883950879, 7.48206244061132668655557186412, 8.162244655544744862092529867049

Graph of the ZZ-function along the critical line