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Properties

Label 2-384-1.1-c1-0-4
Degree 22
Conductor 384384
Sign 11
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
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 + 2·7-s + 9-s + 4·11-s − 6·13-s + 6·17-s + 2·21-s + 4·23-s − 5·25-s + 27-s − 4·29-s + 10·31-s + 4·33-s − 2·37-s − 6·39-s − 2·41-s − 8·43-s − 12·47-s − 3·49-s + 6·51-s + 12·53-s + 4·59-s − 2·61-s + 2·63-s − 4·67-s + 4·69-s − 4·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.436·21-s + 0.834·23-s − 25-s + 0.192·27-s − 0.742·29-s + 1.79·31-s + 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 1.75·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s + 0.520·59-s − 0.256·61-s + 0.251·63-s − 0.488·67-s + 0.481·69-s − 0.474·71-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 11
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 384, ( :1/2), 1)(2,\ 384,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7895659981.789565998
L(12)L(\frac12) ≈\approx 1.7895659981.789565998
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
good5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−11.69179794771208099164944938022, −10.12070860408780938514099192974, −9.634186835262726325802863954007, −8.503024042291174545390595385787, −7.67554775307177371631314255080, −6.80910210319960057293306406499, −5.36858848597467135489376840463, −4.36074490911347519116057298154, −3.06807634409715093203408940942, −1.59108885184574423909298130831, 1.59108885184574423909298130831, 3.06807634409715093203408940942, 4.36074490911347519116057298154, 5.36858848597467135489376840463, 6.80910210319960057293306406499, 7.67554775307177371631314255080, 8.503024042291174545390595385787, 9.634186835262726325802863954007, 10.12070860408780938514099192974, 11.69179794771208099164944938022

Graph of the ZZ-function along the critical line