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Properties

Label 4-1440e2-1.1-c1e2-0-24
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 132.214132.214
Root an. cond. 3.390933.39093
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 6·11-s + 3·17-s + 5·19-s + 25-s − 27-s − 6·33-s + 3·41-s + 5·43-s + 2·49-s − 3·51-s − 5·57-s − 12·59-s + 5·67-s − 14·73-s − 75-s + 81-s − 3·89-s − 2·97-s + 6·99-s + 24·107-s + 3·113-s + 14·121-s − 3·123-s + 127-s − 5·129-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.727·17-s + 1.14·19-s + 1/5·25-s − 0.192·27-s − 1.04·33-s + 0.468·41-s + 0.762·43-s + 2/7·49-s − 0.420·51-s − 0.662·57-s − 1.56·59-s + 0.610·67-s − 1.63·73-s − 0.115·75-s + 1/9·81-s − 0.317·89-s − 0.203·97-s + 0.603·99-s + 2.32·107-s + 0.282·113-s + 1.27·121-s − 0.270·123-s + 0.0887·127-s − 0.440·129-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 132.214132.214
Root analytic conductor: 3.390933.39093
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :1/2,1/2), 1)(4,\ 2073600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3520206822.352020682
L(12)L(\frac12) ≈\approx 2.3520206822.352020682
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
3C1C_1 1+T 1 + T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C2C_2×\timesC2C_2 (1−6T+pT2)(1+pT2) ( 1 - 6 T + p T^{2} )( 1 + p T^{2} )
13C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
19C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C22C_2^2 1+19T2+p2T4 1 + 19 T^{2} + p^{2} T^{4}
29C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
31C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + p^{2} T^{4}
37C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−3T+pT2)(1+pT2) ( 1 - 3 T + p T^{2} )( 1 + p T^{2} )
43C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
47C22C_2^2 1+67T2+p2T4 1 + 67 T^{2} + p^{2} T^{4}
53C22C_2^2 1−56T2+p2T4 1 - 56 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1+3T+pT2)(1+9T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} )
61C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−7T+pT2)(1+2T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} )
71C22C_2^2 1+79T2+p2T4 1 + 79 T^{2} + p^{2} T^{4}
73C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
79C22C_2^2 1+76T2+p2T4 1 + 76 T^{2} + p^{2} T^{4}
83C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−8T+pT2)(1+10T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 10 T + p 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

−7.61619072146803180351107274278, −7.32422567621999186181458748318, −6.85278354150625382127638658286, −6.47309432360768562373488257082, −5.94980494712683988441466278632, −5.75692729802251925801362291311, −5.19925930941201698624733331730, −4.63613235656986934692777973760, −4.26783944619574240034166488210, −3.75420866021664331005836601218, −3.27853030806207598860295079984, −2.77551732002357642454342747808, −1.84126909684414982145330054843, −1.30609692143021771033125671673, −0.73193225387404772207120302717, 0.73193225387404772207120302717, 1.30609692143021771033125671673, 1.84126909684414982145330054843, 2.77551732002357642454342747808, 3.27853030806207598860295079984, 3.75420866021664331005836601218, 4.26783944619574240034166488210, 4.63613235656986934692777973760, 5.19925930941201698624733331730, 5.75692729802251925801362291311, 5.94980494712683988441466278632, 6.47309432360768562373488257082, 6.85278354150625382127638658286, 7.32422567621999186181458748318, 7.61619072146803180351107274278

Graph of the ZZ-function along the critical line