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Properties

Label 2-3840-1.1-c1-0-22
Degree 22
Conductor 38403840
Sign 11
Analytic cond. 30.662530.6625
Root an. cond. 5.537375.53737
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 2·7-s + 9-s + 2·11-s + 2·13-s − 15-s + 2·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 2·33-s − 2·35-s − 2·37-s + 2·39-s + 10·41-s + 12·43-s − 45-s − 4·47-s − 3·49-s + 2·53-s − 2·55-s + 10·59-s − 8·61-s + 2·63-s − 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.269·55-s + 1.30·59-s − 1.02·61-s + 0.251·63-s − 0.248·65-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: 11
Analytic conductor: 30.662530.6625
Root analytic conductor: 5.537375.53737
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3840, ( :1/2), 1)(2,\ 3840,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6470326182.647032618
L(12)L(\frac12) ≈\approx 2.6470326182.647032618
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+pT2 1 - 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−12T+pT2 1 - 12 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p 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

−8.547810659680337642624255862630, −7.76941571723357021542519786506, −7.22945478481046549211272292133, −6.33214957923142631509775977844, −5.47435936704460129444891179504, −4.47683036487743968640759593051, −3.93888758298540112651115143689, −3.02317097146693530017544332839, −1.97182907119826815287047351781, −0.970159482132133752136391193546, 0.970159482132133752136391193546, 1.97182907119826815287047351781, 3.02317097146693530017544332839, 3.93888758298540112651115143689, 4.47683036487743968640759593051, 5.47435936704460129444891179504, 6.33214957923142631509775977844, 7.22945478481046549211272292133, 7.76941571723357021542519786506, 8.547810659680337642624255862630

Graph of the ZZ-function along the critical line