↑

Properties

Label 4-3344e2-1.1-c0e2-0-0
Degree 44
Conductor 1118233611182336
Sign 11
Analytic cond. 2.785132.78513
Root an. cond. 1.291841.29184
Motivic weight 00
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  + 2·5-s − 2·7-s − 2·9-s − 11-s − 2·19-s + 25-s − 4·35-s − 2·43-s − 4·45-s + 49-s − 2·55-s + 4·63-s + 2·77-s + 3·81-s + 4·83-s − 4·95-s + 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2·5-s − 2·7-s − 2·9-s − 11-s − 2·19-s + 25-s − 4·35-s − 2·43-s − 4·45-s + 49-s − 2·55-s + 4·63-s + 2·77-s + 3·81-s + 4·83-s − 4·95-s + 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯

Functional equation

Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1118233611182336    =    28⋅112⋅1922^{8} \cdot 11^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 2.785132.78513
Root analytic conductor: 1.291841.29184
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11182336, ( :0,0), 1)(4,\ 11182336,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.49551637530.4955163753
L(12)L(\frac12) ≈\approx 0.49551637530.4955163753
L(1)L(1) 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
11C2C_2 1+T+T2 1 + T + T^{2}
19C1C_1 (1+T)2 ( 1 + T )^{2}
good3C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
5C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
7C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
37C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
41C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
43C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
71C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
73C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
83C1C_1 (1−T)4 ( 1 - T )^{4}
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
97C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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

−9.116124634607260209711976603861, −8.673920470438124244113198324602, −8.391024482195788646960181817374, −7.936182055181733746265555181945, −7.62851729711186109030202678200, −6.78971483632858839388669619393, −6.42290523941451150081624021416, −6.33940024170226370871815005867, −6.22699194579585236291741971785, −5.52868302880481185877540965669, −5.45524093200267442324615891568, −5.03377124109541591979337953843, −4.45711715415669880473255214204, −3.60501805484142871921563506478, −3.45401214877957642521004217867, −2.87163757491949561897780663283, −2.52618364648759361627890946559, −2.16602627131889975193118298740, −1.75515824237564680179202437865, −0.37325347208455571229472566048, 0.37325347208455571229472566048, 1.75515824237564680179202437865, 2.16602627131889975193118298740, 2.52618364648759361627890946559, 2.87163757491949561897780663283, 3.45401214877957642521004217867, 3.60501805484142871921563506478, 4.45711715415669880473255214204, 5.03377124109541591979337953843, 5.45524093200267442324615891568, 5.52868302880481185877540965669, 6.22699194579585236291741971785, 6.33940024170226370871815005867, 6.42290523941451150081624021416, 6.78971483632858839388669619393, 7.62851729711186109030202678200, 7.936182055181733746265555181945, 8.391024482195788646960181817374, 8.673920470438124244113198324602, 9.116124634607260209711976603861

Graph of the ZZ-function along the critical line