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Properties

Label 2-2368-1.1-c1-0-43
Degree 22
Conductor 23682368
Sign −1-1
Analytic cond. 18.908518.9085
Root an. cond. 4.348394.34839
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 − 7-s − 2·9-s − 3·11-s + 4·13-s + 6·17-s − 2·19-s + 21-s + 6·23-s − 5·25-s + 5·27-s + 6·29-s − 4·31-s + 3·33-s − 37-s − 4·39-s − 9·41-s − 8·43-s + 3·47-s − 6·49-s − 6·51-s + 3·53-s + 2·57-s − 12·59-s − 8·61-s + 2·63-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.218·21-s + 1.25·23-s − 25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.640·39-s − 1.40·41-s − 1.21·43-s + 0.437·47-s − 6/7·49-s − 0.840·51-s + 0.412·53-s + 0.264·57-s − 1.56·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + ⋯

Functional equation

Λ(s)=(2368s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2368s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23682368    =    26⋅372^{6} \cdot 37
Sign: −1-1
Analytic conductor: 18.908518.9085
Root analytic conductor: 4.348394.34839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2368, ( :1/2), −1)(2,\ 2368,\ (\ :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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
37 1+T 1 + T
good3 1+T+pT2 1 + T + p T^{2} 1.3.b
5 1+pT2 1 + p T^{2} 1.5.a
7 1+T+pT2 1 + T + p T^{2} 1.7.b
11 1+3T+pT2 1 + 3 T + p T^{2} 1.11.d
13 1−4T+pT2 1 - 4 T + p T^{2} 1.13.ae
17 1−6T+pT2 1 - 6 T + p T^{2} 1.17.ag
19 1+2T+pT2 1 + 2 T + p T^{2} 1.19.c
23 1−6T+pT2 1 - 6 T + p T^{2} 1.23.ag
29 1−6T+pT2 1 - 6 T + p T^{2} 1.29.ag
31 1+4T+pT2 1 + 4 T + p T^{2} 1.31.e
41 1+9T+pT2 1 + 9 T + p T^{2} 1.41.j
43 1+8T+pT2 1 + 8 T + p T^{2} 1.43.i
47 1−3T+pT2 1 - 3 T + p T^{2} 1.47.ad
53 1−3T+pT2 1 - 3 T + p T^{2} 1.53.ad
59 1+12T+pT2 1 + 12 T + p T^{2} 1.59.m
61 1+8T+pT2 1 + 8 T + p T^{2} 1.61.i
67 1−4T+pT2 1 - 4 T + p T^{2} 1.67.ae
71 1+15T+pT2 1 + 15 T + p T^{2} 1.71.p
73 1−11T+pT2 1 - 11 T + p T^{2} 1.73.al
79 1+10T+pT2 1 + 10 T + p T^{2} 1.79.k
83 1+9T+pT2 1 + 9 T + p T^{2} 1.83.j
89 1−6T+pT2 1 - 6 T + p T^{2} 1.89.ag
97 1−8T+pT2 1 - 8 T + p T^{2} 1.97.ai
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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

−8.487594403486838813155108109695, −7.953080976467459568741615470163, −6.92702526110685059626904983928, −6.13026790010533892031415629049, −5.52397436821147602871559735263, −4.81124860316907390767416903100, −3.49656973901712201010716232454, −2.90414738025441870892963779928, −1.39370194281586028856128276808, 0, 1.39370194281586028856128276808, 2.90414738025441870892963779928, 3.49656973901712201010716232454, 4.81124860316907390767416903100, 5.52397436821147602871559735263, 6.13026790010533892031415629049, 6.92702526110685059626904983928, 7.953080976467459568741615470163, 8.487594403486838813155108109695

Graph of the ZZ-function along the critical line