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Properties

Label 4-778752-1.1-c1e2-0-13
Degree 44
Conductor 778752778752
Sign 11
Analytic cond. 49.653949.6539
Root an. cond. 2.654532.65453
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 9-s − 12·17-s + 8·23-s − 10·25-s − 16·31-s + 4·47-s + 34·49-s − 8·63-s + 4·71-s − 20·73-s + 16·79-s + 81-s − 24·89-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 96·119-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 3.02·7-s + 1/3·9-s − 2.91·17-s + 1.66·23-s − 2·25-s − 2.87·31-s + 0.583·47-s + 34/7·49-s − 1.00·63-s + 0.474·71-s − 2.34·73-s + 1.80·79-s + 1/9·81-s − 2.54·89-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 8.80·119-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(778752s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 778752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(778752s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 778752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 778752778752    =    29⋅32⋅1322^{9} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 49.653949.6539
Root analytic conductor: 2.654532.65453
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 778752, ( :1/2,1/2), 1)(4,\ 778752,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
53C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
59C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{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

−12.9817377283, −12.2699770508, −12.2589845991, −11.3219642411, −11.1129058648, −10.7543074453, −10.3241583179, −9.68560567231, −9.48200090796, −9.23901367889, −8.89738790993, −8.46002418237, −7.59390007236, −7.10423169387, −6.97842775016, −6.43898703961, −6.27116899687, −5.57720303585, −5.24673574244, −4.25093555029, −3.96331013716, −3.60029656187, −2.81789710116, −2.52521406721, −1.67240281366, 0, 0, 1.67240281366, 2.52521406721, 2.81789710116, 3.60029656187, 3.96331013716, 4.25093555029, 5.24673574244, 5.57720303585, 6.27116899687, 6.43898703961, 6.97842775016, 7.10423169387, 7.59390007236, 8.46002418237, 8.89738790993, 9.23901367889, 9.48200090796, 9.68560567231, 10.3241583179, 10.7543074453, 11.1129058648, 11.3219642411, 12.2589845991, 12.2699770508, 12.9817377283

Graph of the ZZ-function along the critical line