↑

Properties

Label 4-2550e2-1.1-c1e2-0-1
Degree 44
Conductor 65025006502500
Sign 11
Analytic cond. 414.605414.605
Root an. cond. 4.512414.51241
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 6·11-s + 16-s + 14·19-s − 12·29-s − 14·31-s + 36-s − 12·41-s + 6·44-s + 13·49-s − 20·61-s − 64-s − 12·71-s − 14·76-s − 34·79-s + 81-s + 24·89-s + 6·99-s − 30·101-s + 14·109-s + 12·116-s + 5·121-s + 14·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.80·11-s + 1/4·16-s + 3.21·19-s − 2.22·29-s − 2.51·31-s + 1/6·36-s − 1.87·41-s + 0.904·44-s + 13/7·49-s − 2.56·61-s − 1/8·64-s − 1.42·71-s − 1.60·76-s − 3.82·79-s + 1/9·81-s + 2.54·89-s + 0.603·99-s − 2.98·101-s + 1.34·109-s + 1.11·116-s + 5/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

Λ(s)=(6502500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6502500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 65025006502500    =    22⋅32⋅54⋅1722^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 414.605414.605
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 6502500, ( :1/2,1/2), 1)(4,\ 6502500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.26861137290.2686113729
L(12)L(\frac12) ≈\approx 0.26861137290.2686113729
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}
3C2C_2 1+T2 1 + T^{2}
5 1 1
17C2C_2 1+T2 1 + T^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
37C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
47C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
53C22C_2^2 1−97T2+p2T4 1 - 97 T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
67C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
71C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2 (1+17T+pT2)2 ( 1 + 17 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
97C22C_2^2 1−94T2+p2T4 1 - 94 T^{2} + p^{2} T^{4}
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

−9.279353336671326402025373360324, −8.756441586066739600777039885811, −8.395875628819618426841910050780, −7.69990400716311455162435486876, −7.52431935486464251488106197899, −7.34587201565838584997364620744, −7.18922747516542450082361005697, −6.21514010011937852838357450134, −5.80541278797801446915087566900, −5.34506079162002874048220604551, −5.32972968541709628887106147473, −5.07838740298404740682706793087, −4.26653256048889989246882328442, −3.82056319352285405482983521168, −3.22297936132339876477210583767, −3.13277066085658569817202219749, −2.51957767955154180736522142661, −1.72632459164441281525344687421, −1.33781476584183833567644085153, −0.17387535493892419288918454133, 0.17387535493892419288918454133, 1.33781476584183833567644085153, 1.72632459164441281525344687421, 2.51957767955154180736522142661, 3.13277066085658569817202219749, 3.22297936132339876477210583767, 3.82056319352285405482983521168, 4.26653256048889989246882328442, 5.07838740298404740682706793087, 5.32972968541709628887106147473, 5.34506079162002874048220604551, 5.80541278797801446915087566900, 6.21514010011937852838357450134, 7.18922747516542450082361005697, 7.34587201565838584997364620744, 7.52431935486464251488106197899, 7.69990400716311455162435486876, 8.395875628819618426841910050780, 8.756441586066739600777039885811, 9.279353336671326402025373360324

Graph of the ZZ-function along the critical line