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Properties

Label 4-3240e2-1.1-c0e2-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5-s + 7-s + 8-s + 10-s − 2·11-s + 13-s − 14-s − 16-s − 2·19-s + 2·22-s + 23-s − 26-s − 35-s + 4·37-s + 2·38-s − 40-s + 41-s − 46-s + 47-s + 49-s − 2·53-s + 2·55-s + 56-s + 59-s + 64-s − 65-s + ⋯
L(s)  = 1  − 2-s − 5-s + 7-s + 8-s + 10-s − 2·11-s + 13-s − 14-s − 16-s − 2·19-s + 2·22-s + 23-s − 26-s − 35-s + 4·37-s + 2·38-s − 40-s + 41-s − 46-s + 47-s + 49-s − 2·53-s + 2·55-s + 56-s + 59-s + 64-s − 65-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 0.56403593550.5640359355
L(12)L(\frac12) ≈\approx 0.56403593550.5640359355
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)
bad2C2C_2 1+T+T2 1 + T + T^{2}
3 1 1
5C2C_2 1+T+T2 1 + T + T^{2}
good7C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
11C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
13C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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} )
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C1C_1 (1−T)4 ( 1 - T )^{4}
41C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
53C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
59C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + 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}
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
89C1C_1 (1−T)4 ( 1 - T )^{4}
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

−8.942633923580969335391689214883, −8.480676807611327247294799801035, −8.209373712572557973251502358896, −8.003579776703579370758019413072, −7.63357820498692989287934724390, −7.58152837765408716159878823074, −7.03238605825532987182737946157, −6.34197433296330745668163290648, −6.10226298989492232937594995415, −5.68749489108613563593076449986, −5.03957871765039858921651706046, −4.65063956843494897380920665463, −4.52720176469718369979864929211, −4.02992299739737853962169585471, −3.62347378956749057768147089426, −2.85468991959902631254159574940, −2.35505118207077451704885693061, −2.10995009594499634705880901142, −1.14257753522792945873190143295, −0.63088498262957786877999365299, 0.63088498262957786877999365299, 1.14257753522792945873190143295, 2.10995009594499634705880901142, 2.35505118207077451704885693061, 2.85468991959902631254159574940, 3.62347378956749057768147089426, 4.02992299739737853962169585471, 4.52720176469718369979864929211, 4.65063956843494897380920665463, 5.03957871765039858921651706046, 5.68749489108613563593076449986, 6.10226298989492232937594995415, 6.34197433296330745668163290648, 7.03238605825532987182737946157, 7.58152837765408716159878823074, 7.63357820498692989287934724390, 8.003579776703579370758019413072, 8.209373712572557973251502358896, 8.480676807611327247294799801035, 8.942633923580969335391689214883

Graph of the ZZ-function along the critical line