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Properties

Label 4-930e2-1.1-c1e2-0-1
Degree 44
Conductor 864900864900
Sign 11
Analytic cond. 55.146755.1467
Root an. cond. 2.725082.72508
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  − 4-s − 2·5-s − 9-s − 10·11-s + 16-s + 10·19-s + 2·20-s − 25-s + 4·29-s + 2·31-s + 36-s + 12·41-s + 10·44-s + 2·45-s + 13·49-s + 20·55-s − 20·59-s − 28·61-s − 64-s − 18·71-s − 10·76-s − 10·79-s − 2·80-s + 81-s − 6·89-s − 20·95-s + 10·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 2.29·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s + 0.359·31-s + 1/6·36-s + 1.87·41-s + 1.50·44-s + 0.298·45-s + 13/7·49-s + 2.69·55-s − 2.60·59-s − 3.58·61-s − 1/8·64-s − 2.13·71-s − 1.14·76-s − 1.12·79-s − 0.223·80-s + 1/9·81-s − 0.635·89-s − 2.05·95-s + 1.00·99-s + ⋯

Functional equation

Λ(s)=(864900s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(864900s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 864900864900    =    22⋅32⋅52⋅3122^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}
Sign: 11
Analytic conductor: 55.146755.1467
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 864900, ( :1/2,1/2), 1)(4,\ 864900,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54670399860.5467039986
L(12)L(\frac12) ≈\approx 0.54670399860.5467039986
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}
3C2C_2 1+T2 1 + T^{2}
5C2C_2 1+2T+pT2 1 + 2 T + p T^{2}
31C1C_1 (1−T)2 ( 1 - T )^{2}
good7C22C_2^2 1−13T2+p2T4 1 - 13 T^{2} + p^{2} T^{4}
11C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
19C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
23C22C_2^2 1+35T2+p2T4 1 + 35 T^{2} + p^{2} T^{4}
29C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−85T2+p2T4 1 - 85 T^{2} + p^{2} T^{4}
47C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
53C22C_2^2 1+63T2+p2T4 1 + 63 T^{2} + p^{2} T^{4}
59C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
61C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
73C22C_2^2 1−65T2+p2T4 1 - 65 T^{2} + p^{2} T^{4}
79C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p 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

−10.18874160642155211997753089142, −10.10600486064744863422335040363, −9.211098191991994807640338601733, −9.192122096228278683362482097397, −8.561884869855235754382523425160, −7.953132079078275551480790536103, −7.64378437401921216055901242366, −7.61616000176016779642244096715, −7.29121855227043982553249199146, −6.22374225769867074162870602737, −5.87805932560752718461395752671, −5.30221655571935217398086244202, −5.19178200871947861699615301855, −4.36839168767893088398497180253, −4.29577267265008237659604437346, −3.12450224960058915117254787891, −2.98857294562736104201925937535, −2.63243592550045901050677890798, −1.40172750430398820767725177859, −0.36339773528994289038468446489, 0.36339773528994289038468446489, 1.40172750430398820767725177859, 2.63243592550045901050677890798, 2.98857294562736104201925937535, 3.12450224960058915117254787891, 4.29577267265008237659604437346, 4.36839168767893088398497180253, 5.19178200871947861699615301855, 5.30221655571935217398086244202, 5.87805932560752718461395752671, 6.22374225769867074162870602737, 7.29121855227043982553249199146, 7.61616000176016779642244096715, 7.64378437401921216055901242366, 7.953132079078275551480790536103, 8.561884869855235754382523425160, 9.192122096228278683362482097397, 9.211098191991994807640338601733, 10.10600486064744863422335040363, 10.18874160642155211997753089142

Graph of the ZZ-function along the critical line