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Properties

Label 2-433-1.1-c1-0-33
Degree 22
Conductor 433433
Sign 11
Analytic cond. 3.457523.45752
Root an. cond. 1.859441.85944
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s − 4-s − 4·5-s + 2·6-s − 3·7-s + 3·8-s + 9-s + 4·10-s − 4·11-s + 2·12-s − 5·13-s + 3·14-s + 8·15-s − 16-s − 3·17-s − 18-s − 4·19-s + 4·20-s + 6·21-s + 4·22-s + 8·23-s − 6·24-s + 11·25-s + 5·26-s + 4·27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s + 0.577·12-s − 1.38·13-s + 0.801·14-s + 2.06·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.894·20-s + 1.30·21-s + 0.852·22-s + 1.66·23-s − 1.22·24-s + 11/5·25-s + 0.980·26-s + 0.769·27-s + 0.566·28-s + ⋯

Functional equation

Λ(s)=(433s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(433s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 433 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 433433
Sign: 11
Analytic conductor: 3.457523.45752
Root analytic conductor: 1.859441.85944
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 433, ( :1/2), 1)(2,\ 433,\ (\ :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)
bad433 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1+3T+pT2 1 + 3 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1+5T+pT2 1 + 5 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1+9T+pT2 1 + 9 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+12T+pT2 1 + 12 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

−10.65914823427479669007317594203, −9.307903179910026089353913014640, −8.432605385178941132054991521013, −7.41239186062132188132348593599, −6.81362847351019760597193961457, −5.16976335052993119549954852181, −4.53754586413642223166086062553, −3.13653467946233451337264151475, 0, 0, 3.13653467946233451337264151475, 4.53754586413642223166086062553, 5.16976335052993119549954852181, 6.81362847351019760597193961457, 7.41239186062132188132348593599, 8.432605385178941132054991521013, 9.307903179910026089353913014640, 10.65914823427479669007317594203

Graph of the ZZ-function along the critical line