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Properties

Label 4-2028e2-1.1-c0e2-0-2
Degree 44
Conductor 41127844112784
Sign 11
Analytic cond. 1.024351.02435
Root an. cond. 1.006031.00603
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 − 2·5-s − 4·8-s − 9-s + 4·10-s + 2·11-s + 5·16-s + 2·18-s − 6·20-s − 4·22-s + 2·25-s − 6·32-s − 3·36-s + 8·40-s + 2·41-s + 6·44-s + 2·45-s + 2·47-s − 4·50-s − 4·55-s + 2·59-s + 7·64-s + 2·71-s + 4·72-s − 10·80-s + 81-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 2·5-s − 4·8-s − 9-s + 4·10-s + 2·11-s + 5·16-s + 2·18-s − 6·20-s − 4·22-s + 2·25-s − 6·32-s − 3·36-s + 8·40-s + 2·41-s + 6·44-s + 2·45-s + 2·47-s − 4·50-s − 4·55-s + 2·59-s + 7·64-s + 2·71-s + 4·72-s − 10·80-s + 81-s + ⋯

Functional equation

Λ(s)=(4112784s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(4112784s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 41127844112784    =    24⋅32⋅1342^{4} \cdot 3^{2} \cdot 13^{4}
Sign: 11
Analytic conductor: 1.024351.02435
Root analytic conductor: 1.006031.00603
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4112784, ( :0,0), 1)(4,\ 4112784,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.33048221120.3304822112
L(12)L(\frac12) ≈\approx 0.33048221120.3304822112
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}
3C2C_2 1+T2 1 + T^{2}
13 1 1
good5C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
7C22C_2^2 1+T4 1 + T^{4}
11C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
17C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
19C22C_2^2 1+T4 1 + T^{4}
23C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C22C_2^2 1+T4 1 + T^{4}
37C22C_2^2 1+T4 1 + T^{4}
41C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
59C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
61C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
67C22C_2^2 1+T4 1 + T^{4}
71C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
73C22C_2^2 1+T4 1 + T^{4}
79C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
83C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
89C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
97C22C_2^2 1+T4 1 + T^{4}
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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

−9.429175970839669992314016320493, −8.844411680549665924755941856157, −8.788311079053535587231515379599, −8.466576564368633891575538956094, −8.152524119260823843322354538919, −7.49459206035581429032282318774, −7.45100208269666008629506502182, −7.08787643542006332680443736993, −6.47245280326769663830592376429, −6.38367174198439832934284263040, −5.65647216135132157282707735115, −5.45313388208230079270890482121, −4.41201995288830945696042840231, −3.91394188591685008002032078162, −3.79452005981392926041927245158, −3.12557753267479880337052821483, −2.64911719924869438233920884333, −2.07769305522838036469947486518, −1.14134639604706353862457351261, −0.68281756649636679151848371362, 0.68281756649636679151848371362, 1.14134639604706353862457351261, 2.07769305522838036469947486518, 2.64911719924869438233920884333, 3.12557753267479880337052821483, 3.79452005981392926041927245158, 3.91394188591685008002032078162, 4.41201995288830945696042840231, 5.45313388208230079270890482121, 5.65647216135132157282707735115, 6.38367174198439832934284263040, 6.47245280326769663830592376429, 7.08787643542006332680443736993, 7.45100208269666008629506502182, 7.49459206035581429032282318774, 8.152524119260823843322354538919, 8.466576564368633891575538956094, 8.788311079053535587231515379599, 8.844411680549665924755941856157, 9.429175970839669992314016320493

Graph of the ZZ-function along the critical line