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Properties

Label 4-20e2-1.1-c2e2-0-0
Degree 44
Conductor 400400
Sign 11
Analytic cond. 0.2969810.296981
Root an. cond. 0.7382140.738214
Motivic weight 22
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  − 4·4-s + 6·5-s − 18·9-s + 16·16-s − 24·20-s + 11·25-s + 84·29-s + 72·36-s − 36·41-s − 108·45-s − 98·49-s + 44·61-s − 64·64-s + 96·80-s + 243·81-s − 156·89-s − 44·100-s − 396·101-s + 364·109-s − 336·116-s + 242·121-s − 84·125-s + 127-s + 131-s + 137-s + 139-s − 288·144-s + ⋯
L(s)  = 1  − 4-s + 6/5·5-s − 2·9-s + 16-s − 6/5·20-s + 0.439·25-s + 2.89·29-s + 2·36-s − 0.878·41-s − 2.39·45-s − 2·49-s + 0.721·61-s − 64-s + 6/5·80-s + 3·81-s − 1.75·89-s − 0.439·100-s − 3.92·101-s + 3.33·109-s − 2.89·116-s + 2·121-s − 0.671·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2·144-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 0.2969810.296981
Root analytic conductor: 0.7382140.738214
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 400, ( :1,1), 1)(4,\ 400,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.68717845780.6871784578
L(12)L(\frac12) ≈\approx 0.68717845780.6871784578
L(2)L(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)
bad2C2C_2 1+p2T2 1 + p^{2} T^{2}
5C2C_2 1−6T+p2T2 1 - 6 T + p^{2} T^{2}
good3C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
7C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
11C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
13C2C_2 (1−10T+p2T2)(1+10T+p2T2) ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} )
17C2C_2 (1−30T+p2T2)(1+30T+p2T2) ( 1 - 30 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} )
19C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
23C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
29C2C_2 (1−42T+p2T2)2 ( 1 - 42 T + p^{2} T^{2} )^{2}
31C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
37C2C_2 (1−70T+p2T2)(1+70T+p2T2) ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} )
41C2C_2 (1+18T+p2T2)2 ( 1 + 18 T + p^{2} T^{2} )^{2}
43C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
47C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
53C2C_2 (1−90T+p2T2)(1+90T+p2T2) ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} )
59C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
61C2C_2 (1−22T+p2T2)2 ( 1 - 22 T + p^{2} T^{2} )^{2}
67C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
71C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
73C2C_2 (1−110T+p2T2)(1+110T+p2T2) ( 1 - 110 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} )
79C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
83C2C_2 (1+p2T2)2 ( 1 + p^{2} T^{2} )^{2}
89C2C_2 (1+78T+p2T2)2 ( 1 + 78 T + p^{2} T^{2} )^{2}
97C2C_2 (1−130T+p2T2)(1+130T+p2T2) ( 1 - 130 T + p^{2} T^{2} )( 1 + 130 T + p^{2} 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

−17.96062590567325815104023523697, −17.94885027651970621775195863569, −17.33399256142956227684504519776, −16.87855723835633671836689974348, −16.12067171012368066821763565845, −15.02155578445027214922818563134, −14.34275859722948862111176454258, −13.91615602008035128526283404749, −13.59782577234706309083810554246, −12.62713949989021404985482988469, −11.90550244298984676775472499594, −11.02542479449272167448992573251, −10.09372891876765178042217088505, −9.547024869041151195264697437329, −8.520031008385668286043288259253, −8.324070771202873362775595250013, −6.50419717571286079875050761495, −5.70955971770895197519786291105, −4.87586661189760822228494762830, −2.98280952584571453573837281815, 2.98280952584571453573837281815, 4.87586661189760822228494762830, 5.70955971770895197519786291105, 6.50419717571286079875050761495, 8.324070771202873362775595250013, 8.520031008385668286043288259253, 9.547024869041151195264697437329, 10.09372891876765178042217088505, 11.02542479449272167448992573251, 11.90550244298984676775472499594, 12.62713949989021404985482988469, 13.59782577234706309083810554246, 13.91615602008035128526283404749, 14.34275859722948862111176454258, 15.02155578445027214922818563134, 16.12067171012368066821763565845, 16.87855723835633671836689974348, 17.33399256142956227684504519776, 17.94885027651970621775195863569, 17.96062590567325815104023523697

Graph of the ZZ-function along the critical line