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Properties

Label 4-450e2-1.1-c3e2-0-11
Degree 44
Conductor 202500202500
Sign 11
Analytic cond. 704.948704.948
Root an. cond. 5.152755.15275
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  + 4·2-s + 12·4-s + 32·8-s + 80·16-s + 124·17-s + 168·19-s + 280·23-s + 32·31-s + 192·32-s + 496·34-s + 672·38-s + 1.12e3·46-s + 200·47-s − 190·49-s + 1.47e3·53-s − 716·61-s + 128·62-s + 448·64-s + 1.48e3·68-s + 2.01e3·76-s − 1.87e3·79-s + 2.60e3·83-s + 3.36e3·92-s + 800·94-s − 760·98-s + 5.90e3·106-s + 1.39e3·107-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 1.76·17-s + 2.02·19-s + 2.53·23-s + 0.185·31-s + 1.06·32-s + 2.50·34-s + 2.86·38-s + 3.58·46-s + 0.620·47-s − 0.553·49-s + 3.82·53-s − 1.50·61-s + 0.262·62-s + 7/8·64-s + 2.65·68-s + 3.04·76-s − 2.66·79-s + 3.44·83-s + 3.80·92-s + 0.877·94-s − 0.783·98-s + 5.40·106-s + 1.25·107-s + ⋯

Functional equation

Λ(s)=(202500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(202500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 202500202500    =    22⋅34⋅542^{2} \cdot 3^{4} \cdot 5^{4}
Sign: 11
Analytic conductor: 704.948704.948
Root analytic conductor: 5.152755.15275
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 202500, ( :3/2,3/2), 1)(4,\ 202500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 10.4944417010.49444170
L(12)L(\frac12) ≈\approx 10.4944417010.49444170
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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
3 1 1
5 1 1
good7C22C_2^2 1+190T2+p6T4 1 + 190 T^{2} + p^{6} T^{4}
11C22C_2^2 1+2166T2+p6T4 1 + 2166 T^{2} + p^{6} T^{4}
13C22C_2^2 1−70T2+p6T4 1 - 70 T^{2} + p^{6} T^{4}
17C2C_2 (1−62T+p3T2)2 ( 1 - 62 T + p^{3} T^{2} )^{2}
19C2C_2 (1−84T+p3T2)2 ( 1 - 84 T + p^{3} T^{2} )^{2}
23C2C_2 (1−140T+p3T2)2 ( 1 - 140 T + p^{3} T^{2} )^{2}
29C22C_2^2 1+8602T2+p6T4 1 + 8602 T^{2} + p^{6} T^{4}
31C2C_2 (1−16T+p3T2)2 ( 1 - 16 T + p^{3} T^{2} )^{2}
37C22C_2^2 1+41290T2+p6T4 1 + 41290 T^{2} + p^{6} T^{4}
41C22C_2^2 1+88242T2+p6T4 1 + 88242 T^{2} + p^{6} T^{4}
43C22C_2^2 1+32038T2+p6T4 1 + 32038 T^{2} + p^{6} T^{4}
47C2C_2 (1−100T+p3T2)2 ( 1 - 100 T + p^{3} T^{2} )^{2}
53C2C_2 (1−738T+p3T2)2 ( 1 - 738 T + p^{3} T^{2} )^{2}
59C22C_2^2 1−6378T2+p6T4 1 - 6378 T^{2} + p^{6} T^{4}
61C2C_2 (1+358T+p3T2)2 ( 1 + 358 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−114698T2+p6T4 1 - 114698 T^{2} + p^{6} T^{4}
71C22C_2^2 1−159122T2+p6T4 1 - 159122 T^{2} + p^{6} T^{4}
73C22C_2^2 1+579634T2+p6T4 1 + 579634 T^{2} + p^{6} T^{4}
79C2C_2 (1+936T+p3T2)2 ( 1 + 936 T + p^{3} T^{2} )^{2}
83C2C_2 (1−1304T+p3T2)2 ( 1 - 1304 T + p^{3} T^{2} )^{2}
89C22C_2^2 1+902034T2+p6T4 1 + 902034 T^{2} + p^{6} T^{4}
97C22C_2^2 1+1251970T2+p6T4 1 + 1251970 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

−10.91325510212989642997366542315, −10.68718432887346367326811546760, −9.949048989581920647197480122687, −9.855758807957044587102758869553, −8.900879207728493049964041196471, −8.877068859602440315619710431198, −7.78759338580560266958756108244, −7.57343723924811754597840143077, −7.16902440058342457029628776373, −6.72075832164600314028619023790, −5.95557537195315445080649149502, −5.58037578033031578540529797356, −5.01034348210061453900338334788, −4.96465329895456523770127855772, −3.88596975781471889340555873069, −3.53573356912624628836371954998, −2.92567263138210704051459502448, −2.55143500753069135015772712354, −1.22000691428943146410391995160, −1.02322585538005756882312502793, 1.02322585538005756882312502793, 1.22000691428943146410391995160, 2.55143500753069135015772712354, 2.92567263138210704051459502448, 3.53573356912624628836371954998, 3.88596975781471889340555873069, 4.96465329895456523770127855772, 5.01034348210061453900338334788, 5.58037578033031578540529797356, 5.95557537195315445080649149502, 6.72075832164600314028619023790, 7.16902440058342457029628776373, 7.57343723924811754597840143077, 7.78759338580560266958756108244, 8.877068859602440315619710431198, 8.900879207728493049964041196471, 9.855758807957044587102758869553, 9.949048989581920647197480122687, 10.68718432887346367326811546760, 10.91325510212989642997366542315

Graph of the ZZ-function along the critical line