↑

Properties

Label 4-395136-1.1-c1e2-0-14
Degree 44
Conductor 395136395136
Sign 11
Analytic cond. 25.194225.1942
Root an. cond. 2.240392.24039
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 9-s + 12·19-s + 6·25-s − 4·29-s + 8·37-s + 12·47-s + 49-s − 8·59-s + 63-s + 81-s − 20·83-s + 4·103-s − 24·109-s + 4·113-s − 6·121-s + 127-s + 131-s + 12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯
L(s)  = 1  + 0.377·7-s + 1/3·9-s + 2.75·19-s + 6/5·25-s − 0.742·29-s + 1.31·37-s + 1.75·47-s + 1/7·49-s − 1.04·59-s + 0.125·63-s + 1/9·81-s − 2.19·83-s + 0.394·103-s − 2.29·109-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 1.04·133-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 − 1.38·169-s + ⋯

Functional equation

Λ(s)=(395136s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(395136s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 395136395136    =    27⋅32⋅732^{7} \cdot 3^{2} \cdot 7^{3}
Sign: 11
Analytic conductor: 25.194225.1942
Root analytic conductor: 2.240392.24039
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 395136, ( :1/2,1/2), 1)(4,\ 395136,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2945314322.294531432
L(12)L(\frac12) ≈\approx 2.2945314322.294531432
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
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
7C1C_1 1−T 1 - T
good5C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2×\timesC2C_2 (1−8T+pT2)(1−4T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} )
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1−10T+pT2)(1+2T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
67C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
71C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
73C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
79C22C_2^2 1−114T2+p2T4 1 - 114 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1+8T+pT2)(1+12T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
97C22C_2^2 1−106T2+p2T4 1 - 106 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

−8.817092243723242181725296380715, −8.080160915792226774767951985221, −7.58740369991958940622336029269, −7.44250648293099723920733236326, −6.94413503049518840685816373067, −6.32902796163253772483999548071, −5.70926449015034340385417075013, −5.35198542114396338677619100144, −4.90490602070295855727773873402, −4.26150609384425412337007274683, −3.75686516057931722112806716393, −2.98575273566609576233473549037, −2.64553871551281105656158994906, −1.52382382536395764588885012083, −0.944589603320793487264556311363, 0.944589603320793487264556311363, 1.52382382536395764588885012083, 2.64553871551281105656158994906, 2.98575273566609576233473549037, 3.75686516057931722112806716393, 4.26150609384425412337007274683, 4.90490602070295855727773873402, 5.35198542114396338677619100144, 5.70926449015034340385417075013, 6.32902796163253772483999548071, 6.94413503049518840685816373067, 7.44250648293099723920733236326, 7.58740369991958940622336029269, 8.080160915792226774767951985221, 8.817092243723242181725296380715

Graph of the ZZ-function along the critical line