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Properties

Label 4-2550e2-1.1-c1e2-0-42
Degree 44
Conductor 65025006502500
Sign 11
Analytic cond. 414.605414.605
Root an. cond. 4.512414.51241
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  + 2·3-s − 4-s + 2·7-s + 3·9-s − 2·12-s + 16-s − 2·17-s + 6·19-s + 4·21-s + 16·23-s + 4·27-s − 2·28-s − 3·36-s + 6·37-s + 2·48-s − 11·49-s − 4·51-s + 12·57-s + 16·59-s + 6·63-s − 64-s + 2·68-s + 32·69-s + 8·73-s − 6·76-s + 5·81-s − 4·84-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 0.755·7-s + 9-s − 0.577·12-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.872·21-s + 3.33·23-s + 0.769·27-s − 0.377·28-s − 1/2·36-s + 0.986·37-s + 0.288·48-s − 1.57·49-s − 0.560·51-s + 1.58·57-s + 2.08·59-s + 0.755·63-s − 1/8·64-s + 0.242·68-s + 3.85·69-s + 0.936·73-s − 0.688·76-s + 5/9·81-s − 0.436·84-s + ⋯

Functional equation

Λ(s)=(6502500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6502500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 65025006502500    =    22⋅32⋅54⋅1722^{2} \cdot 3^{2} \cdot 5^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 414.605414.605
Root analytic conductor: 4.512414.51241
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 6502500, ( :1/2,1/2), 1)(4,\ 6502500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.3927107575.392710757
L(12)L(\frac12) ≈\approx 5.3927107575.392710757
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)
bad2C2C_2 1+T2 1 + T^{2}
3C1C_1 (1−T)2 ( 1 - T )^{2}
5 1 1
17C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
good7C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
11C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
19C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
29C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
31C22C_2^2 1−61T2+p2T4 1 - 61 T^{2} + p^{2} T^{4}
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C22C_2^2 1−78T2+p2T4 1 - 78 T^{2} + p^{2} T^{4}
43C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
47C22C_2^2 1+75T2+p2T4 1 + 75 T^{2} + p^{2} T^{4}
53C22C_2^2 1+63T2+p2T4 1 + 63 T^{2} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C22C_2^2 1−125T2+p2T4 1 - 125 T^{2} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
79C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−12T+pT2)2 ( 1 - 12 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

−8.953437266547120230324070130189, −8.775038061632553088223337416682, −8.417509340731736207069433502406, −8.012896242087041538564959686228, −7.54880436723633880549307497293, −7.37034497738504992623997348425, −6.88917404654469952423559197959, −6.62221251103740744320377410909, −5.93949519215987153309427941768, −5.46151443502371642760214485258, −4.92105947126811246238590867589, −4.77784645658636268850378098354, −4.49267728916769438680480260447, −3.67583275624564914238352384063, −3.26642484880613977787182075083, −3.15008615969782863125410887926, −2.37226175374957258653872878470, −2.00348976563689881533517063952, −1.04396355304095248302138119646, −0.936087930532685525070541061256, 0.936087930532685525070541061256, 1.04396355304095248302138119646, 2.00348976563689881533517063952, 2.37226175374957258653872878470, 3.15008615969782863125410887926, 3.26642484880613977787182075083, 3.67583275624564914238352384063, 4.49267728916769438680480260447, 4.77784645658636268850378098354, 4.92105947126811246238590867589, 5.46151443502371642760214485258, 5.93949519215987153309427941768, 6.62221251103740744320377410909, 6.88917404654469952423559197959, 7.37034497738504992623997348425, 7.54880436723633880549307497293, 8.012896242087041538564959686228, 8.417509340731736207069433502406, 8.775038061632553088223337416682, 8.953437266547120230324070130189

Graph of the ZZ-function along the critical line