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Properties

Label 4-3240e2-1.1-c1e2-0-14
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
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  + 5-s − 2·7-s + 11-s − 13-s + 2·17-s + 8·19-s + 23-s − 5·29-s − 31-s − 2·35-s + 12·37-s − 7·43-s + 7·47-s + 7·49-s + 24·53-s + 55-s − 4·59-s − 10·61-s − 65-s + 4·67-s − 24·71-s + 12·73-s − 2·77-s − 15·79-s + 2·83-s + 2·85-s + 24·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s + 0.301·11-s − 0.277·13-s + 0.485·17-s + 1.83·19-s + 0.208·23-s − 0.928·29-s − 0.179·31-s − 0.338·35-s + 1.97·37-s − 1.06·43-s + 1.02·47-s + 49-s + 3.29·53-s + 0.134·55-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s − 2.84·71-s + 1.40·73-s − 0.227·77-s − 1.68·79-s + 0.219·83-s + 0.216·85-s + 2.54·89-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{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: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7412639872.741263987
L(12)L(\frac12) ≈\approx 2.7412639872.741263987
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
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C22C_2^2 1+2T−3T2+2pT3+p2T4 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−T−10T2−pT3+p2T4 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4}
13C22C_2^2 1+T−12T2+pT3+p2T4 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4}
17C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−T−22T2−pT3+p2T4 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4}
29C22C_2^2 1+5T−4T2+5pT3+p2T4 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+T−30T2+pT3+p2T4 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C22C_2^2 1+7T+6T2+7pT3+p2T4 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−7T+2T2−7pT3+p2T4 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4}
53C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+10T+39T2+10pT3+p2T4 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
79C22C_2^2 1+15T+146T2+15pT3+p2T4 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−2T−79T2−2pT3+p2T4 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4}
89C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
97C22C_2^2 1+10T+3T2+10pT3+p2T4 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} 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

−8.978996037818273424428764250109, −8.540223972768996793226948795706, −8.030931578743356619696315012254, −7.59762911680978690345969509947, −7.31804891056379523800901404089, −7.13283953037680506822708484828, −6.55536425793309974024182979886, −6.19879515558044651203530969823, −5.71957074898778087340546010500, −5.47748566771949301448685266056, −5.25117483585517765213115052153, −4.55833196703981417968026599152, −4.03304808304314484724377918909, −3.85022534545496910170789524940, −3.13818904541381699451621627970, −2.86739079588172408049261016459, −2.44945930128364634824539687876, −1.71498687479888353872801320952, −1.14665309631874332067934706220, −0.56611576397386358411542041566, 0.56611576397386358411542041566, 1.14665309631874332067934706220, 1.71498687479888353872801320952, 2.44945930128364634824539687876, 2.86739079588172408049261016459, 3.13818904541381699451621627970, 3.85022534545496910170789524940, 4.03304808304314484724377918909, 4.55833196703981417968026599152, 5.25117483585517765213115052153, 5.47748566771949301448685266056, 5.71957074898778087340546010500, 6.19879515558044651203530969823, 6.55536425793309974024182979886, 7.13283953037680506822708484828, 7.31804891056379523800901404089, 7.59762911680978690345969509947, 8.030931578743356619696315012254, 8.540223972768996793226948795706, 8.978996037818273424428764250109

Graph of the ZZ-function along the critical line