↑

Properties

Label 4-700e2-1.1-c1e2-0-5
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 31.242831.2428
Root an. cond. 2.364212.36421
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  + 6·9-s − 10·11-s + 12·19-s + 6·29-s + 4·31-s − 8·41-s − 49-s + 28·59-s + 8·61-s − 26·71-s − 2·79-s + 27·81-s − 20·89-s − 60·99-s + 24·101-s − 18·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + ⋯
L(s)  = 1  + 2·9-s − 3.01·11-s + 2.75·19-s + 1.11·29-s + 0.718·31-s − 1.24·41-s − 1/7·49-s + 3.64·59-s + 1.02·61-s − 3.08·71-s − 0.225·79-s + 3·81-s − 2.11·89-s − 6.03·99-s + 2.38·101-s − 1.72·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 31.242831.2428
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :1/2,1/2), 1)(4,\ 490000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9949345651.994934565
L(12)L(\frac12) ≈\approx 1.9949345651.994934565
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)
bad2 1 1
5 1 1
7C2C_2 1+T2 1 + T^{2}
good3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C2C_2 (1+5T+pT2)2 ( 1 + 5 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} )
17C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
19C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
23C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
29C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
41C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
43C22C_2^2 1−37T2+p2T4 1 - 37 T^{2} + p^{2} T^{4}
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
59C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
61C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C2C_2 (1+13T+pT2)2 ( 1 + 13 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+T+pT2)2 ( 1 + T + p T^{2} )^{2}
83C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
89C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
97C22C_2^2 1−190T2+p2T4 1 - 190 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

−10.39630201846709653688252958820, −10.10741351579027436891973055389, −9.898435165927122424821339861481, −9.833678509456435057677641150799, −8.880575765166552278343172025796, −8.396994660749056155037365720496, −7.986143326869911592311018799088, −7.56363556880996454325480902808, −7.11519169640077653996075052143, −7.10228689012684436915297422111, −6.22512540844802711355733326436, −5.49812720141374618627204639874, −5.06162779077396476297767527010, −5.05039575756716157407313005772, −4.26109177807554697867478453567, −3.61816846242806993735719031260, −2.86740551482613276423128265527, −2.60098483233759492199007751435, −1.60049408733384802633414554803, −0.790133013805134540606127223683, 0.790133013805134540606127223683, 1.60049408733384802633414554803, 2.60098483233759492199007751435, 2.86740551482613276423128265527, 3.61816846242806993735719031260, 4.26109177807554697867478453567, 5.05039575756716157407313005772, 5.06162779077396476297767527010, 5.49812720141374618627204639874, 6.22512540844802711355733326436, 7.10228689012684436915297422111, 7.11519169640077653996075052143, 7.56363556880996454325480902808, 7.986143326869911592311018799088, 8.396994660749056155037365720496, 8.880575765166552278343172025796, 9.833678509456435057677641150799, 9.898435165927122424821339861481, 10.10741351579027436891973055389, 10.39630201846709653688252958820

Graph of the ZZ-function along the critical line