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Properties

Label 4-350e2-1.1-c3e2-0-3
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 426.450426.450
Root an. cond. 4.544304.54430
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 46·9-s + 18·11-s + 16·16-s + 20·19-s − 378·29-s − 464·31-s + 184·36-s − 876·41-s − 72·44-s − 49·49-s + 1.34e3·59-s + 412·61-s − 64·64-s − 942·71-s − 80·76-s − 1.48e3·79-s + 1.38e3·81-s − 360·89-s − 828·99-s − 1.45e3·101-s + 1.38e3·109-s + 1.51e3·116-s − 2.41e3·121-s + 1.85e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.70·9-s + 0.493·11-s + 1/4·16-s + 0.241·19-s − 2.42·29-s − 2.68·31-s + 0.851·36-s − 3.33·41-s − 0.246·44-s − 1/7·49-s + 2.96·59-s + 0.864·61-s − 1/8·64-s − 1.57·71-s − 0.120·76-s − 2.11·79-s + 1.90·81-s − 0.428·89-s − 0.840·99-s − 1.43·101-s + 1.21·109-s + 1.21·116-s − 1.81·121-s + 1.34·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 426.450426.450
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :3/2,3/2), 1)(4,\ 122500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.094245911430.09424591143
L(12)L(\frac12) ≈\approx 0.094245911430.09424591143
L(52)L(\frac{5}{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}
5 1 1
7C2C_2 1+p2T2 1 + p^{2} T^{2}
good3C22C_2^2 1+46T2+p6T4 1 + 46 T^{2} + p^{6} T^{4}
11C2C_2 (1−9T+p3T2)2 ( 1 - 9 T + p^{3} T^{2} )^{2}
13C2C_2 (1−6pT+p3T2)(1+6pT+p3T2) ( 1 - 6 p T + p^{3} T^{2} )( 1 + 6 p T + p^{3} T^{2} )
17C22C_2^2 1−610T2+p6T4 1 - 610 T^{2} + p^{6} T^{4}
19C2C_2 (1−10T+p3T2)2 ( 1 - 10 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−18709T2+p6T4 1 - 18709 T^{2} + p^{6} T^{4}
29C2C_2 (1+189T+p3T2)2 ( 1 + 189 T + p^{3} T^{2} )^{2}
31C2C_2 (1+232T+p3T2)2 ( 1 + 232 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−8281T2+p6T4 1 - 8281 T^{2} + p^{6} T^{4}
41C2C_2 (1+438T+p3T2)2 ( 1 + 438 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−34405T2+p6T4 1 - 34405 T^{2} + p^{6} T^{4}
47C22C_2^2 1+28550T2+p6T4 1 + 28550 T^{2} + p^{6} T^{4}
53C22C_2^2 1−172438T2+p6T4 1 - 172438 T^{2} + p^{6} T^{4}
59C2C_2 (1−672T+p3T2)2 ( 1 - 672 T + p^{3} T^{2} )^{2}
61C2C_2 (1−206T+p3T2)2 ( 1 - 206 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−242725T2+p6T4 1 - 242725 T^{2} + p^{6} T^{4}
71C2C_2 (1+471T+p3T2)2 ( 1 + 471 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−401038T2+p6T4 1 - 401038 T^{2} + p^{6} T^{4}
79C2C_2 (1+743T+p3T2)2 ( 1 + 743 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−22p2T2+p6T4 1 - 22 p^{2} T^{2} + p^{6} T^{4}
89C2C_2 (1+180T+p3T2)2 ( 1 + 180 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1791490T2+p6T4 1 - 1791490 T^{2} + p^{6} 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

−11.66192988503986676094214094391, −10.85739241904669680892210353223, −10.44394703519454479390961041763, −9.769174214227922185839992031918, −9.397529289028950408330502653531, −8.932920847735023648826039712272, −8.455863350980625372804025968261, −8.333809676569906029628408376742, −7.26627884511919658844955868587, −7.25307312667974695041238543022, −6.44710302245667124470664540980, −5.67259986714009165677218911052, −5.44139187315020638334192370373, −5.14252532814717964385580122860, −4.01036463542792585896394670172, −3.65092981487146002651057922090, −3.13735781912030899113490882638, −2.17553701446599698400587380651, −1.51419165046458572686692626278, −0.10630053046025347886711834822, 0.10630053046025347886711834822, 1.51419165046458572686692626278, 2.17553701446599698400587380651, 3.13735781912030899113490882638, 3.65092981487146002651057922090, 4.01036463542792585896394670172, 5.14252532814717964385580122860, 5.44139187315020638334192370373, 5.67259986714009165677218911052, 6.44710302245667124470664540980, 7.25307312667974695041238543022, 7.26627884511919658844955868587, 8.333809676569906029628408376742, 8.455863350980625372804025968261, 8.932920847735023648826039712272, 9.397529289028950408330502653531, 9.769174214227922185839992031918, 10.44394703519454479390961041763, 10.85739241904669680892210353223, 11.66192988503986676094214094391

Graph of the ZZ-function along the critical line