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Properties

Label 4-450e2-1.1-c3e2-0-3
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  − 4·4-s − 84·11-s + 16·16-s + 230·19-s − 420·29-s − 386·31-s − 24·41-s + 336·44-s + 685·49-s + 1.38e3·59-s − 1.46e3·61-s − 64·64-s + 456·71-s − 920·76-s + 320·79-s − 480·89-s − 1.82e3·101-s + 3.47e3·109-s + 1.68e3·116-s + 2.63e3·121-s + 1.54e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.30·11-s + 1/4·16-s + 2.77·19-s − 2.68·29-s − 2.23·31-s − 0.0914·41-s + 1.15·44-s + 1.99·49-s + 3.04·59-s − 3.07·61-s − 1/8·64-s + 0.762·71-s − 1.38·76-s + 0.455·79-s − 0.571·89-s − 1.79·101-s + 3.04·109-s + 1.34·116-s + 1.97·121-s + 1.11·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-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 0.69316679520.6931667952
L(12)L(\frac12) ≈\approx 0.69316679520.6931667952
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}
3 1 1
5 1 1
good7C22C_2^2 1−685T2+p6T4 1 - 685 T^{2} + p^{6} T^{4}
11C2C_2 (1+42T+p3T2)2 ( 1 + 42 T + p^{3} T^{2} )^{2}
13C22C_2^2 1+95T2+p6T4 1 + 95 T^{2} + p^{6} T^{4}
17C22C_2^2 1−6910T2+p6T4 1 - 6910 T^{2} + p^{6} T^{4}
19C2C_2 (1−115T+p3T2)2 ( 1 - 115 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+1910T2+p6T4 1 + 1910 T^{2} + p^{6} T^{4}
29C2C_2 (1+210T+p3T2)2 ( 1 + 210 T + p^{3} T^{2} )^{2}
31C2C_2 (1+193T+p3T2)2 ( 1 + 193 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−19510T2+p6T4 1 - 19510 T^{2} + p^{6} T^{4}
41C2C_2 (1+12T+p3T2)2 ( 1 + 12 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−89845T2+p6T4 1 - 89845 T^{2} + p^{6} T^{4}
47C22C_2^2 1−36250T2+p6T4 1 - 36250 T^{2} + p^{6} T^{4}
53C22C_2^2 1−260890T2+p6T4 1 - 260890 T^{2} + p^{6} T^{4}
59C2C_2 (1−690T+p3T2)2 ( 1 - 690 T + p^{3} T^{2} )^{2}
61C2C_2 (1+733T+p3T2)2 ( 1 + 733 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−512125T2+p6T4 1 - 512125 T^{2} + p^{6} T^{4}
71C2C_2 (1−228T+p3T2)2 ( 1 - 228 T + p^{3} T^{2} )^{2}
73C22C_2^2 1+101810T2+p6T4 1 + 101810 T^{2} + p^{6} T^{4}
79C2C_2 (1−160T+p3T2)2 ( 1 - 160 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−930130T2+p6T4 1 - 930130 T^{2} + p^{6} T^{4}
89C2C_2 (1+240T+p3T2)2 ( 1 + 240 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1564225T2+p6T4 1 - 1564225 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

−10.83873681475319718413332433348, −10.42358364976021129169946794179, −10.04631242273619391487746896641, −9.402775764559550739188966706582, −9.276509670919285682082372819313, −8.739591927397558823403244219767, −8.004239565543575578434790469912, −7.52123912447999566358368013415, −7.48998783759318748165935589343, −7.03886836608338327275433017618, −5.72425587523346107784076344787, −5.71360486807984569434128470588, −5.20584491168773341484575028387, −4.93138856214640589143034796965, −3.73581182558786197460205492013, −3.65225842048490600129525126996, −2.78991282878349325069165891270, −2.20996701283792806053974829948, −1.28978331279074381337272596413, −0.28117832935911331330552939217, 0.28117832935911331330552939217, 1.28978331279074381337272596413, 2.20996701283792806053974829948, 2.78991282878349325069165891270, 3.65225842048490600129525126996, 3.73581182558786197460205492013, 4.93138856214640589143034796965, 5.20584491168773341484575028387, 5.71360486807984569434128470588, 5.72425587523346107784076344787, 7.03886836608338327275433017618, 7.48998783759318748165935589343, 7.52123912447999566358368013415, 8.004239565543575578434790469912, 8.739591927397558823403244219767, 9.276509670919285682082372819313, 9.402775764559550739188966706582, 10.04631242273619391487746896641, 10.42358364976021129169946794179, 10.83873681475319718413332433348

Graph of the ZZ-function along the critical line