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Properties

Label 2-240-1.1-c3-0-11
Degree 22
Conductor 240240
Sign −1-1
Analytic cond. 14.160414.1604
Root an. cond. 3.763033.76303
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  + 3·3-s + 5·5-s − 32·7-s + 9·9-s − 36·11-s − 10·13-s + 15·15-s − 78·17-s − 140·19-s − 96·21-s + 192·23-s + 25·25-s + 27·27-s + 6·29-s + 16·31-s − 108·33-s − 160·35-s − 34·37-s − 30·39-s − 390·41-s + 52·43-s + 45·45-s − 408·47-s + 681·49-s − 234·51-s − 114·53-s − 180·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.72·7-s + 1/3·9-s − 0.986·11-s − 0.213·13-s + 0.258·15-s − 1.11·17-s − 1.69·19-s − 0.997·21-s + 1.74·23-s + 1/5·25-s + 0.192·27-s + 0.0384·29-s + 0.0926·31-s − 0.569·33-s − 0.772·35-s − 0.151·37-s − 0.123·39-s − 1.48·41-s + 0.184·43-s + 0.149·45-s − 1.26·47-s + 1.98·49-s − 0.642·51-s − 0.295·53-s − 0.441·55-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: −1-1
Analytic conductor: 14.160414.1604
Root analytic conductor: 3.763033.76303
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 240, ( :3/2), −1)(2,\ 240,\ (\ :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)
bad2 1 1
3 1−pT 1 - p T
5 1−pT 1 - p T
good7 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
11 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
13 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
17 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
19 1+140T+p3T2 1 + 140 T + p^{3} T^{2}
23 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
29 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
31 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
37 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
41 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
43 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
47 1+408T+p3T2 1 + 408 T + p^{3} T^{2}
53 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
59 1+516T+p3T2 1 + 516 T + p^{3} T^{2}
61 1+58T+p3T2 1 + 58 T + p^{3} T^{2}
67 1−892T+p3T2 1 - 892 T + p^{3} T^{2}
71 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
73 1+646T+p3T2 1 + 646 T + p^{3} T^{2}
79 1−1168T+p3T2 1 - 1168 T + p^{3} T^{2}
83 1−732T+p3T2 1 - 732 T + p^{3} T^{2}
89 1+1590T+p3T2 1 + 1590 T + p^{3} T^{2}
97 1−2pT+p3T2 1 - 2 p 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

−10.93938158024673929899604813785, −10.14220325816311825207945120638, −9.269499407560692814971699815077, −8.462446752671038454783541104852, −6.98318920215576826243112993680, −6.32707470388400166271658294227, −4.82924913449240888347252086771, −3.31443828568712893748331786799, −2.33530647153277444876327397949, 0, 2.33530647153277444876327397949, 3.31443828568712893748331786799, 4.82924913449240888347252086771, 6.32707470388400166271658294227, 6.98318920215576826243112993680, 8.462446752671038454783541104852, 9.269499407560692814971699815077, 10.14220325816311825207945120638, 10.93938158024673929899604813785

Graph of the ZZ-function along the critical line