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Properties

Label 4-150e2-1.1-c7e2-0-1
Degree 44
Conductor 2250022500
Sign 11
Analytic cond. 2195.642195.64
Root an. cond. 6.845276.84527
Motivic weight 77
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  − 64·4-s − 729·9-s − 120·11-s + 4.09e3·16-s − 1.07e5·19-s + 1.77e5·29-s − 2.40e5·31-s + 4.66e4·36-s − 1.02e6·41-s + 7.68e3·44-s + 1.59e6·49-s + 3.77e6·59-s − 5.43e5·61-s − 2.62e5·64-s − 5.01e6·71-s + 6.85e6·76-s + 1.47e7·79-s + 5.31e5·81-s − 1.13e7·89-s + 8.74e4·99-s + 3.09e7·101-s − 2.79e7·109-s − 1.13e7·116-s − 3.89e7·121-s + 1.54e7·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 0.0271·11-s + 1/4·16-s − 3.58·19-s + 1.34·29-s − 1.45·31-s + 1/6·36-s − 2.31·41-s + 0.0135·44-s + 1.93·49-s + 2.39·59-s − 0.306·61-s − 1/8·64-s − 1.66·71-s + 1.79·76-s + 3.35·79-s + 1/9·81-s − 1.70·89-s + 0.00906·99-s + 2.98·101-s − 2.06·109-s − 0.674·116-s − 1.99·121-s + 0.725·124-s + ⋯

Functional equation

Λ(s)=(22500s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(22500s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2250022500    =    22⋅32⋅542^{2} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 2195.642195.64
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 22500, ( :7/2,7/2), 1)(4,\ 22500,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.045114864120.04511486412
L(12)L(\frac12) ≈\approx 0.045114864120.04511486412
L(92)L(\frac{9}{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+p6T2 1 + p^{6} T^{2}
3C2C_2 1+p6T2 1 + p^{6} T^{2}
5 1 1
good7C22C_2^2 1−1593262T2+p14T4 1 - 1593262 T^{2} + p^{14} T^{4}
11C2C_2 (1+60T+p7T2)2 ( 1 + 60 T + p^{7} T^{2} )^{2}
13C22C_2^2 1+72017882T2+p14T4 1 + 72017882 T^{2} + p^{14} T^{4}
17C22C_2^2 1−626409502T2+p14T4 1 - 626409502 T^{2} + p^{14} T^{4}
19C2C_2 (1+53564T+p7T2)2 ( 1 + 53564 T + p^{7} T^{2} )^{2}
23C22C_2^2 1−6080650894T2+p14T4 1 - 6080650894 T^{2} + p^{14} T^{4}
29C2C_2 (1−88554T+p7T2)2 ( 1 - 88554 T + p^{7} T^{2} )^{2}
31C2C_2 (1+120352T+p7T2)2 ( 1 + 120352 T + p^{7} T^{2} )^{2}
37C22C_2^2 1+30703611050T2+p14T4 1 + 30703611050 T^{2} + p^{14} T^{4}
41C2C_2 (1+510246T+p7T2)2 ( 1 + 510246 T + p^{7} T^{2} )^{2}
43C22C_2^2 1−522492572470T2+p14T4 1 - 522492572470 T^{2} + p^{14} T^{4}
47C22C_2^2 1−416699614p2T2+p14T4 1 - 416699614 p^{2} T^{2} + p^{14} T^{4}
53C22C_2^2 1−2340511297270T2+p14T4 1 - 2340511297270 T^{2} + p^{14} T^{4}
59C2C_2 (1−1885740T+p7T2)2 ( 1 - 1885740 T + p^{7} T^{2} )^{2}
61C2C_2 (1+271690T+p7T2)2 ( 1 + 271690 T + p^{7} T^{2} )^{2}
67C22C_2^2 1−10966511900422T2+p14T4 1 - 10966511900422 T^{2} + p^{14} T^{4}
71C2C_2 (1+2505480T+p7T2)2 ( 1 + 2505480 T + p^{7} T^{2} )^{2}
73C22C_2^2 1−13365336789838T2+p14T4 1 - 13365336789838 T^{2} + p^{14} T^{4}
79C2C_2 (1−7354768T+p7T2)2 ( 1 - 7354768 T + p^{7} T^{2} )^{2}
83C22C_2^2 1−39474570255718T2+p14T4 1 - 39474570255718 T^{2} + p^{14} T^{4}
89C2C_2 (1+5685162T+p7T2)2 ( 1 + 5685162 T + p^{7} T^{2} )^{2}
97C22C_2^2 1−97894684984510T2+p14T4 1 - 97894684984510 T^{2} + p^{14} 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

−11.98278587301169367229765565053, −11.47377464616022301050744821196, −10.74783962696890619061605247948, −10.36971350999318702306159097420, −10.18855390185646514100500907918, −9.173077592368046417606235682767, −8.643659432951926488431297602800, −8.608561267472088070993384310798, −7.914274395369269704113067026260, −7.10502438244692577174599222356, −6.48708723227243605919253497021, −6.18347946927772991782764127424, −5.26387778019591580998147894678, −4.83698977910186591106787327839, −3.95050769453963841946478366774, −3.76985407135732910749375646218, −2.52199166568518671393580452347, −2.15007180517046887256470830912, −1.18153489576986631782379282170, −0.06064438411723357706532854012, 0.06064438411723357706532854012, 1.18153489576986631782379282170, 2.15007180517046887256470830912, 2.52199166568518671393580452347, 3.76985407135732910749375646218, 3.95050769453963841946478366774, 4.83698977910186591106787327839, 5.26387778019591580998147894678, 6.18347946927772991782764127424, 6.48708723227243605919253497021, 7.10502438244692577174599222356, 7.914274395369269704113067026260, 8.608561267472088070993384310798, 8.643659432951926488431297602800, 9.173077592368046417606235682767, 10.18855390185646514100500907918, 10.36971350999318702306159097420, 10.74783962696890619061605247948, 11.47377464616022301050744821196, 11.98278587301169367229765565053

Graph of the ZZ-function along the critical line