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Properties

Label 2-384-1.1-c1-0-6
Degree 22
Conductor 384384
Sign −1-1
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 11

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: −1-1
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: 11
Selberg data: (2, 384, ( :1/2), −1)(2,\ 384,\ (\ :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
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

−10.72157176858504266602545238224, −10.03012842475637444340101123739, −9.355375973689939441990781827382, −7.73185283482156320096864018539, −7.29744951786402483785793531451, −5.81271121311664380914911812156, −5.25073137830748218691985779847, −3.77490371552045331982722842815, −2.36547685273488160999702672858, 0, 2.36547685273488160999702672858, 3.77490371552045331982722842815, 5.25073137830748218691985779847, 5.81271121311664380914911812156, 7.29744951786402483785793531451, 7.73185283482156320096864018539, 9.355375973689939441990781827382, 10.03012842475637444340101123739, 10.72157176858504266602545238224

Graph of the ZZ-function along the critical line