↑

Properties

Label 4-810e2-1.1-c1e2-0-23
Degree 44
Conductor 656100656100
Sign 11
Analytic cond. 41.833541.8335
Root an. cond. 2.543202.54320
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·5-s − 4·11-s + 16-s − 12·19-s + 2·20-s − 25-s − 18·29-s − 4·31-s − 22·41-s + 4·44-s + 13·49-s + 8·55-s + 8·59-s − 14·61-s − 64-s − 12·71-s + 12·76-s + 24·79-s − 2·80-s − 2·89-s + 24·95-s + 100-s + 4·101-s − 14·109-s + 18·116-s − 10·121-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1.20·11-s + 1/4·16-s − 2.75·19-s + 0.447·20-s − 1/5·25-s − 3.34·29-s − 0.718·31-s − 3.43·41-s + 0.603·44-s + 13/7·49-s + 1.07·55-s + 1.04·59-s − 1.79·61-s − 1/8·64-s − 1.42·71-s + 1.37·76-s + 2.70·79-s − 0.223·80-s − 0.211·89-s + 2.46·95-s + 1/10·100-s + 0.398·101-s − 1.34·109-s + 1.67·116-s − 0.909·121-s + ⋯

Functional equation

Λ(s)=(656100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(656100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 656100656100    =    22⋅38⋅522^{2} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 41.833541.8335
Root analytic conductor: 2.543202.54320
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 656100, ( :1/2,1/2), 1)(4,\ 656100,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1+T2 1 + T^{2}
3 1 1
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
19C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
23C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
29C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
31C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
67C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
89C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
97C2C_2 (1−18T+pT2)(1+18T+pT2) ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} )
show more
show less
   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.17883907488857588694831212536, −9.586506805980223172737277751495, −8.927895939250911386391102622991, −8.924112324095231111501790289339, −8.193778223586007174045619339017, −8.090914453733625316899062029754, −7.38424231783881445130406252641, −7.28747656041900917098478454085, −6.54201123385877517129664791782, −6.10521963207707746450126980464, −5.35628086966803970879463285667, −5.29633122825214424114841254839, −4.52829153612623436996189583037, −3.95206760983480762547768841836, −3.78168104628012741309522551527, −3.10427971779357953583827368712, −2.11786822605608749364722969462, −1.86627265350834450046512039765, 0, 0, 1.86627265350834450046512039765, 2.11786822605608749364722969462, 3.10427971779357953583827368712, 3.78168104628012741309522551527, 3.95206760983480762547768841836, 4.52829153612623436996189583037, 5.29633122825214424114841254839, 5.35628086966803970879463285667, 6.10521963207707746450126980464, 6.54201123385877517129664791782, 7.28747656041900917098478454085, 7.38424231783881445130406252641, 8.090914453733625316899062029754, 8.193778223586007174045619339017, 8.924112324095231111501790289339, 8.927895939250911386391102622991, 9.586506805980223172737277751495, 10.17883907488857588694831212536

Graph of the ZZ-function along the critical line