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Properties

Label 4-4050e2-1.1-c1e2-0-34
Degree 44
Conductor 1640250016402500
Sign 11
Analytic cond. 1045.831045.83
Root an. cond. 5.686775.68677
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 4·13-s − 8·14-s + 5·16-s − 2·19-s − 6·23-s − 8·26-s − 12·28-s + 4·31-s + 6·32-s − 16·37-s − 4·38-s − 4·43-s − 12·46-s − 6·47-s + 49-s − 12·52-s − 18·53-s − 16·56-s + 4·61-s + 8·62-s + 7·64-s − 16·67-s + 12·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 1.10·13-s − 2.13·14-s + 5/4·16-s − 0.458·19-s − 1.25·23-s − 1.56·26-s − 2.26·28-s + 0.718·31-s + 1.06·32-s − 2.63·37-s − 0.648·38-s − 0.609·43-s − 1.76·46-s − 0.875·47-s + 1/7·49-s − 1.66·52-s − 2.47·53-s − 2.13·56-s + 0.512·61-s + 1.01·62-s + 7/8·64-s − 1.95·67-s + 1.42·71-s + ⋯

Functional equation

Λ(s)=(16402500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(16402500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1640250016402500    =    22⋅38⋅542^{2} \cdot 3^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1045.831045.83
Root analytic conductor: 5.686775.68677
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 16402500, ( :1/2,1/2), 1)(4,\ 16402500,\ (\ :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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
5 1 1
good7D4D_{4} 1+4T+15T2+4pT3+p2T4 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13D4D_{4} 1+4T+27T2+4pT3+p2T4 1 + 4 T + 27 T^{2} + 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
19D4D_{4} 1+2T−9T2+2pT3+p2T4 1 + 2 T - 9 T^{2} + 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+6T+43T2+6pT3+p2T4 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
31D4D_{4} 1−4T+54T2−4pT3+p2T4 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4}
37C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
41C22C_2^2 1+55T2+p2T4 1 + 55 T^{2} + p^{2} T^{4}
43D4D_{4} 1+4T+78T2+4pT3+p2T4 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1+6T+91T2+6pT3+p2T4 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+18T+175T2+18pT3+p2T4 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+43T2+p2T4 1 + 43 T^{2} + p^{2} T^{4}
61D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
67C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+8T+66T2+8pT3+p2T4 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+24T+298T2+24pT3+p2T4 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4}
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97D4D_{4} 1+16T+210T2+16pT3+p2T4 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4}
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   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

−8.016948402030064565863561924130, −7.930634814744980147377337556239, −7.28316109088332046334786287334, −6.99697371067983458645161584759, −6.58212775219323659447922185535, −6.49082823515108118429248150928, −6.08115872466775017042183637807, −5.56386009620794927755614690211, −5.26347936668625295831886516238, −4.92071088805352290561763617906, −4.29280623860668206962206143488, −4.23010252821686291171883985852, −3.47343377950167578428961548105, −3.34933732230151683333544520629, −2.78919757863597125203275196827, −2.58180466540315450577668951260, −1.69868254046163333588646506862, −1.59137209604272726220549827532, 0, 0, 1.59137209604272726220549827532, 1.69868254046163333588646506862, 2.58180466540315450577668951260, 2.78919757863597125203275196827, 3.34933732230151683333544520629, 3.47343377950167578428961548105, 4.23010252821686291171883985852, 4.29280623860668206962206143488, 4.92071088805352290561763617906, 5.26347936668625295831886516238, 5.56386009620794927755614690211, 6.08115872466775017042183637807, 6.49082823515108118429248150928, 6.58212775219323659447922185535, 6.99697371067983458645161584759, 7.28316109088332046334786287334, 7.930634814744980147377337556239, 8.016948402030064565863561924130

Graph of the ZZ-function along the critical line