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Properties

Label 4-3240e2-1.1-c0e2-0-17
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 2.614592.61459
Root an. cond. 1.271601.27160
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 2.614592.61459
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :0,0), 1)(4,\ 10497600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 7.2790797427.279079742
L(12)L(\frac12) ≈\approx 7.2790797427.279079742
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
11C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
13C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
23C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
53C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
59C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
61C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
83C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
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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

−8.985415510924040841652380932550, −8.644431530835748219528419054636, −8.013046953223404996845233618761, −7.71548024687354504338870134728, −7.22920137302163859514935722425, −7.19342726863504151283290926730, −6.58598200328578019443890668890, −6.27324337157929223761397343478, −5.75645420659034615353133038412, −5.57426773100597854767857035521, −5.15994669509305035617800269702, −5.13865282203846834763251167716, −4.25294712941792186571947289940, −4.02295480227100309543703328782, −3.55573528832098950261198487490, −3.12388743892840018489691919905, −2.74207081252684412975609324357, −2.15725181972387401659466000115, −1.56808751671950202533415454776, −1.46879081297093408809818655709, 1.46879081297093408809818655709, 1.56808751671950202533415454776, 2.15725181972387401659466000115, 2.74207081252684412975609324357, 3.12388743892840018489691919905, 3.55573528832098950261198487490, 4.02295480227100309543703328782, 4.25294712941792186571947289940, 5.13865282203846834763251167716, 5.15994669509305035617800269702, 5.57426773100597854767857035521, 5.75645420659034615353133038412, 6.27324337157929223761397343478, 6.58598200328578019443890668890, 7.19342726863504151283290926730, 7.22920137302163859514935722425, 7.71548024687354504338870134728, 8.013046953223404996845233618761, 8.644431530835748219528419054636, 8.985415510924040841652380932550

Graph of the ZZ-function along the critical line