Properties

Label 4-220e2-1.1-c3e2-0-0
Degree 44
Conductor 4840048400
Sign 11
Analytic cond. 168.491168.491
Root an. cond. 3.602833.60283
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  − 5·5-s + 35·9-s − 22·11-s + 136·19-s − 100·25-s − 520·29-s + 350·31-s − 760·41-s − 175·45-s + 610·49-s + 110·55-s + 286·59-s + 1.35e3·61-s + 2.07e3·71-s − 436·79-s + 496·81-s − 2.55e3·89-s − 680·95-s − 770·99-s + 1.27e3·101-s + 284·109-s + 363·121-s + 1.12e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.29·9-s − 0.603·11-s + 1.64·19-s − 4/5·25-s − 3.32·29-s + 2.02·31-s − 2.89·41-s − 0.579·45-s + 1.77·49-s + 0.269·55-s + 0.631·59-s + 2.83·61-s + 3.46·71-s − 0.620·79-s + 0.680·81-s − 3.04·89-s − 0.734·95-s − 0.781·99-s + 1.25·101-s + 0.249·109-s + 3/11·121-s + 0.804·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

Λ(s)=(48400s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(48400s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4840048400    =    24521122^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 168.491168.491
Root analytic conductor: 3.602833.60283
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 48400, ( :3/2,3/2), 1)(4,\ 48400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 2.0354277552.035427755
L(12)L(\frac12) \approx 2.0354277552.035427755
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
5C2C_2 1+pT+p3T2 1 + p T + p^{3} T^{2}
11C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3C22C_2^2 135T2+p6T4 1 - 35 T^{2} + p^{6} T^{4}
7C2C_2 (136T+p3T2)(1+36T+p3T2) ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} )
13C22C_2^2 1+470T2+p6T4 1 + 470 T^{2} + p^{6} T^{4}
17C22C_2^2 19142T2+p6T4 1 - 9142 T^{2} + p^{6} T^{4}
19C2C_2 (168T+p3T2)2 ( 1 - 68 T + p^{3} T^{2} )^{2}
23C22C_2^2 110483T2+p6T4 1 - 10483 T^{2} + p^{6} T^{4}
29C2C_2 (1+260T+p3T2)2 ( 1 + 260 T + p^{3} T^{2} )^{2}
31C2C_2 (1175T+p3T2)2 ( 1 - 175 T + p^{3} T^{2} )^{2}
37C22C_2^2 172407T2+p6T4 1 - 72407 T^{2} + p^{6} T^{4}
41C2C_2 (1+380T+p3T2)2 ( 1 + 380 T + p^{3} T^{2} )^{2}
43C22C_2^2 165914T2+p6T4 1 - 65914 T^{2} + p^{6} T^{4}
47C22C_2^2 1114546T2+p6T4 1 - 114546 T^{2} + p^{6} T^{4}
53C22C_2^2 192250T2+p6T4 1 - 92250 T^{2} + p^{6} T^{4}
59C2C_2 (1143T+p3T2)2 ( 1 - 143 T + p^{3} T^{2} )^{2}
61C2C_2 (1676T+p3T2)2 ( 1 - 676 T + p^{3} T^{2} )^{2}
67C22C_2^2 1323347T2+p6T4 1 - 323347 T^{2} + p^{6} T^{4}
71C2C_2 (11035T+p3T2)2 ( 1 - 1035 T + p^{3} T^{2} )^{2}
73C22C_2^2 1668290T2+p6T4 1 - 668290 T^{2} + p^{6} T^{4}
79C2C_2 (1+218T+p3T2)2 ( 1 + 218 T + p^{3} T^{2} )^{2}
83C22C_2^2 1568330T2+p6T4 1 - 568330 T^{2} + p^{6} T^{4}
89C2C_2 (1+1279T+p3T2)2 ( 1 + 1279 T + p^{3} T^{2} )^{2}
97C22C_2^2 11230095T2+p6T4 1 - 1230095 T^{2} + p^{6} T^{4}
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   L(s)=p j=14(1αj,pps)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

−11.95074739280655888122812547081, −11.58639949252771081875024192292, −11.30284408933867571540769753388, −10.56376511680353018762772406797, −9.897713089129453427350787820338, −9.859277426808712231543834332675, −9.347860888262493976750365016387, −8.376785139584930230905656127960, −8.184744302757415598630857206314, −7.43236208204780404458438339723, −7.12897220904777658051349220430, −6.71616314121717887024652794718, −5.58230306706859417250584546049, −5.42010855150368838168576024934, −4.66051454224359381589732379373, −3.71069324037631876847393722610, −3.66923080155452296311962486922, −2.43516102729658480829965200191, −1.63036085856316864720049048689, −0.61625561810630477085879647291, 0.61625561810630477085879647291, 1.63036085856316864720049048689, 2.43516102729658480829965200191, 3.66923080155452296311962486922, 3.71069324037631876847393722610, 4.66051454224359381589732379373, 5.42010855150368838168576024934, 5.58230306706859417250584546049, 6.71616314121717887024652794718, 7.12897220904777658051349220430, 7.43236208204780404458438339723, 8.184744302757415598630857206314, 8.376785139584930230905656127960, 9.347860888262493976750365016387, 9.859277426808712231543834332675, 9.897713089129453427350787820338, 10.56376511680353018762772406797, 11.30284408933867571540769753388, 11.58639949252771081875024192292, 11.95074739280655888122812547081

Graph of the ZZ-function along the critical line