Properties

Label 4-175e2-1.1-c9e2-0-1
Degree 44
Conductor 3062530625
Sign 11
Analytic cond. 8123.648123.64
Root an. cond. 9.493749.49374
Motivic weight 99
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  + 6·2-s + 86·3-s − 804·4-s + 516·6-s + 4.80e3·7-s − 6.79e3·8-s − 1.04e4·9-s + 3.53e4·11-s − 6.91e4·12-s + 2.65e4·13-s + 2.88e4·14-s + 3.90e5·16-s + 4.63e5·17-s − 6.27e4·18-s − 9.25e5·19-s + 4.12e5·21-s + 2.11e5·22-s − 7.78e5·23-s − 5.84e5·24-s + 1.59e5·26-s − 7.43e5·27-s − 3.86e6·28-s − 1.00e7·29-s + 2.46e6·31-s + 4.00e6·32-s + 3.03e6·33-s + 2.78e6·34-s + ⋯
L(s)  = 1  + 0.265·2-s + 0.612·3-s − 1.57·4-s + 0.162·6-s + 0.755·7-s − 0.586·8-s − 0.531·9-s + 0.727·11-s − 0.962·12-s + 0.257·13-s + 0.200·14-s + 1.49·16-s + 1.34·17-s − 0.140·18-s − 1.62·19-s + 0.463·21-s + 0.192·22-s − 0.579·23-s − 0.359·24-s + 0.0683·26-s − 0.269·27-s − 1.18·28-s − 2.62·29-s + 0.479·31-s + 0.675·32-s + 0.445·33-s + 0.357·34-s + ⋯

Functional equation

Λ(s)=(30625s/2ΓC(s)2L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(30625s/2ΓC(s+9/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3062530625    =    54725^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 8123.648123.64
Root analytic conductor: 9.493749.49374
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 30625, ( :9/2,9/2), 1)(4,\ 30625,\ (\ :9/2, 9/2),\ 1)

Particular Values

L(5)L(5) \approx 1.8709757651.870975765
L(12)L(\frac12) \approx 1.8709757651.870975765
L(112)L(\frac{11}{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)
bad5 1 1
7C1C_1 (1p4T)2 ( 1 - p^{4} T )^{2}
good2D4D_{4} 13pT+105p3T23p10T3+p18T4 1 - 3 p T + 105 p^{3} T^{2} - 3 p^{10} T^{3} + p^{18} T^{4}
3D4D_{4} 186T+5954pT286p9T3+p18T4 1 - 86 T + 5954 p T^{2} - 86 p^{9} T^{3} + p^{18} T^{4}
11D4D_{4} 135316T+2892681078T235316p9T3+p18T4 1 - 35316 T + 2892681078 T^{2} - 35316 p^{9} T^{3} + p^{18} T^{4}
13D4D_{4} 126530T1541163822T226530p9T3+p18T4 1 - 26530 T - 1541163822 T^{2} - 26530 p^{9} T^{3} + p^{18} T^{4}
17D4D_{4} 1463920T+273833245726T2463920p9T3+p18T4 1 - 463920 T + 273833245726 T^{2} - 463920 p^{9} T^{3} + p^{18} T^{4}
19D4D_{4} 1+925426T+858791487510T2+925426p9T3+p18T4 1 + 925426 T + 858791487510 T^{2} + 925426 p^{9} T^{3} + p^{18} T^{4}
23D4D_{4} 1+778128T+3473691840430T2+778128p9T3+p18T4 1 + 778128 T + 3473691840430 T^{2} + 778128 p^{9} T^{3} + p^{18} T^{4}
29D4D_{4} 1+10003584T+52302031706070T2+10003584p9T3+p18T4 1 + 10003584 T + 52302031706070 T^{2} + 10003584 p^{9} T^{3} + p^{18} T^{4}
31D4D_{4} 12467260T+49371575832542T22467260p9T3+p18T4 1 - 2467260 T + 49371575832542 T^{2} - 2467260 p^{9} T^{3} + p^{18} T^{4}
37D4D_{4} 1+30735552T+484209298874630T2+30735552p9T3+p18T4 1 + 30735552 T + 484209298874630 T^{2} + 30735552 p^{9} T^{3} + p^{18} T^{4}
41D4D_{4} 1+19103448T+602984827739166T2+19103448p9T3+p18T4 1 + 19103448 T + 602984827739166 T^{2} + 19103448 p^{9} T^{3} + p^{18} T^{4}
43D4D_{4} 1+4065100T+797231337676374T2+4065100p9T3+p18T4 1 + 4065100 T + 797231337676374 T^{2} + 4065100 p^{9} T^{3} + p^{18} T^{4}
47D4D_{4} 182195020T+3721245520696702T282195020p9T3+p18T4 1 - 82195020 T + 3721245520696702 T^{2} - 82195020 p^{9} T^{3} + p^{18} T^{4}
53D4D_{4} 155189812T+3123778356606670T255189812p9T3+p18T4 1 - 55189812 T + 3123778356606670 T^{2} - 55189812 p^{9} T^{3} + p^{18} T^{4}
59D4D_{4} 1+7069218T+16866494212382134T2+7069218p9T3+p18T4 1 + 7069218 T + 16866494212382134 T^{2} + 7069218 p^{9} T^{3} + p^{18} T^{4}
61D4D_{4} 144316386T+21654336818123658T244316386p9T3+p18T4 1 - 44316386 T + 21654336818123658 T^{2} - 44316386 p^{9} T^{3} + p^{18} T^{4}
67D4D_{4} 1241921336T+59516583718815510T2241921336p9T3+p18T4 1 - 241921336 T + 59516583718815510 T^{2} - 241921336 p^{9} T^{3} + p^{18} T^{4}
71D4D_{4} 1206493816T+58491352612128526T2206493816p9T3+p18T4 1 - 206493816 T + 58491352612128526 T^{2} - 206493816 p^{9} T^{3} + p^{18} T^{4}
73D4D_{4} 1499153188T+178474458263805254T2499153188p9T3+p18T4 1 - 499153188 T + 178474458263805254 T^{2} - 499153188 p^{9} T^{3} + p^{18} T^{4}
79D4D_{4} 15930824pT+239633073722978334T25930824p10T3+p18T4 1 - 5930824 p T + 239633073722978334 T^{2} - 5930824 p^{10} T^{3} + p^{18} T^{4}
83D4D_{4} 1+444023958T+333438010641681622T2+444023958p9T3+p18T4 1 + 444023958 T + 333438010641681622 T^{2} + 444023958 p^{9} T^{3} + p^{18} T^{4}
89D4D_{4} 1636267396T+801539802340191990T2636267396p9T3+p18T4 1 - 636267396 T + 801539802340191990 T^{2} - 636267396 p^{9} T^{3} + p^{18} T^{4}
97D4D_{4} 11632716064T+2180562419544849758T21632716064p9T3+p18T4 1 - 1632716064 T + 2180562419544849758 T^{2} - 1632716064 p^{9} T^{3} + p^{18} 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

−11.24736840267355414168373815103, −10.73513528579450487671986428164, −10.19767358347092717573159112406, −9.681064071062166126534867535311, −9.053738390617711546116856195206, −8.736858325823682805074417413058, −8.467463836465438712973521192822, −7.920675264502913579887568848730, −7.34633947525959988117123588851, −6.57245112476261471319148351636, −5.81019747511315724849956059286, −5.32453252970888275250377022748, −5.00044365436248585046422647362, −4.08115409205300848253758962368, −3.60850188643369384816150837197, −3.59541895133901368702273437857, −2.24912981254981469504134128471, −1.86943577280585272761878224638, −0.954087512157474385978369047606, −0.34002778790331420961708499884, 0.34002778790331420961708499884, 0.954087512157474385978369047606, 1.86943577280585272761878224638, 2.24912981254981469504134128471, 3.59541895133901368702273437857, 3.60850188643369384816150837197, 4.08115409205300848253758962368, 5.00044365436248585046422647362, 5.32453252970888275250377022748, 5.81019747511315724849956059286, 6.57245112476261471319148351636, 7.34633947525959988117123588851, 7.920675264502913579887568848730, 8.467463836465438712973521192822, 8.736858325823682805074417413058, 9.053738390617711546116856195206, 9.681064071062166126534867535311, 10.19767358347092717573159112406, 10.73513528579450487671986428164, 11.24736840267355414168373815103

Graph of the ZZ-function along the critical line