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Properties

Label 2-320892-1.1-c1-0-5
Degree 22
Conductor 320892320892
Sign 11
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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 − 2·5-s − 7-s + 9-s − 13-s + 2·15-s + 17-s − 4·19-s + 21-s − 4·23-s − 25-s − 27-s + 5·29-s + 7·31-s + 2·35-s + 9·37-s + 39-s − 8·41-s − 7·43-s − 2·45-s + 7·47-s − 6·49-s − 51-s − 4·53-s + 4·57-s + 7·59-s + 14·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.928·29-s + 1.25·31-s + 0.338·35-s + 1.47·37-s + 0.160·39-s − 1.24·41-s − 1.06·43-s − 0.298·45-s + 1.02·47-s − 6/7·49-s − 0.140·51-s − 0.549·53-s + 0.529·57-s + 0.911·59-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 320892, ( :1/2), 1)(2,\ 320892,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3928163641.392816364
L(12)L(\frac12) ≈\approx 1.3928163641.392816364
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
11 1 1
13 1+T 1 + T
17 1−T 1 - T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−10T+pT2 1 - 10 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

−12.56503923266172, −12.01988934747962, −11.73889742399802, −11.46259229589040, −10.82016138774006, −10.32925536093423, −9.914096218757320, −9.671195502223396, −8.861730424467769, −8.339837656261442, −8.030400988367099, −7.634967721879849, −6.890295894750609, −6.562167229237568, −6.186755329442824, −5.613368128642556, −4.840823231725882, −4.693703482429881, −3.923188289012181, −3.689012320048927, −2.953016962593401, −2.339283635881637, −1.767025694480862, −0.7875332566447591, −0.4442452632019937, 0.4442452632019937, 0.7875332566447591, 1.767025694480862, 2.339283635881637, 2.953016962593401, 3.689012320048927, 3.923188289012181, 4.693703482429881, 4.840823231725882, 5.613368128642556, 6.186755329442824, 6.562167229237568, 6.890295894750609, 7.634967721879849, 8.030400988367099, 8.339837656261442, 8.861730424467769, 9.671195502223396, 9.914096218757320, 10.32925536093423, 10.82016138774006, 11.46259229589040, 11.73889742399802, 12.01988934747962, 12.56503923266172

Graph of the ZZ-function along the critical line