Properties

Label 4-2205e2-1.1-c3e2-0-3
Degree 44
Conductor 48620254862025
Sign 11
Analytic cond. 16925.816925.8
Root an. cond. 11.406111.4061
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  − 3·2-s + 4-s − 10·5-s − 3·8-s + 30·10-s − 62·11-s + 6·13-s + 9·16-s + 40·17-s + 122·19-s − 10·20-s + 186·22-s − 16·23-s + 75·25-s − 18·26-s − 352·29-s − 66·31-s + 165·32-s − 120·34-s − 188·37-s − 366·38-s + 30·40-s + 16·41-s − 396·43-s − 62·44-s + 48·46-s − 188·47-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s − 0.894·5-s − 0.132·8-s + 0.948·10-s − 1.69·11-s + 0.128·13-s + 9/64·16-s + 0.570·17-s + 1.47·19-s − 0.111·20-s + 1.80·22-s − 0.145·23-s + 3/5·25-s − 0.135·26-s − 2.25·29-s − 0.382·31-s + 0.911·32-s − 0.605·34-s − 0.835·37-s − 1.56·38-s + 0.118·40-s + 0.0609·41-s − 1.40·43-s − 0.212·44-s + 0.153·46-s − 0.583·47-s + ⋯

Functional equation

Λ(s)=(4862025s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(4862025s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 48620254862025    =    3452743^{4} \cdot 5^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 16925.816925.8
Root analytic conductor: 11.406111.4061
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4862025, ( :3/2,3/2), 1)(4,\ 4862025,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 0.59022384940.5902238494
L(12)L(\frac12) \approx 0.59022384940.5902238494
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)
bad3 1 1
5C1C_1 (1+pT)2 ( 1 + p T )^{2}
7 1 1
good2D4D_{4} 1+3T+p3T2+3p3T3+p6T4 1 + 3 T + p^{3} T^{2} + 3 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+62T+3582T2+62p3T3+p6T4 1 + 62 T + 3582 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 16T+3378T26p3T3+p6T4 1 - 6 T + 3378 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 140T+8750T240p3T3+p6T4 1 - 40 T + 8750 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1122T+17398T2122p3T3+p6T4 1 - 122 T + 17398 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+16T+34pT2+16p3T3+p6T4 1 + 16 T + 34 p T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+352T+78278T2+352p3T3+p6T4 1 + 352 T + 78278 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+66T+45870T2+66p3T3+p6T4 1 + 66 T + 45870 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+188T+44542T2+188p3T3+p6T4 1 + 188 T + 44542 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 116T+18350T216p3T3+p6T4 1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+396T+2226pT2+396p3T3+p6T4 1 + 396 T + 2226 p T^{2} + 396 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+4pT+15582T2+4p4T3+p6T4 1 + 4 p T + 15582 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}
53D4D_{4} 1+982T+504354T2+982p3T3+p6T4 1 + 982 T + 504354 T^{2} + 982 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1516T+418118T2516p3T3+p6T4 1 - 516 T + 418118 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1880T+575238T2880p3T3+p6T4 1 - 880 T + 575238 T^{2} - 880 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+356T+100374T2+356p3T3+p6T4 1 + 356 T + 100374 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+310T+664038T2+310p3T3+p6T4 1 + 310 T + 664038 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+326T+789802T2+326p3T3+p6T4 1 + 326 T + 789802 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 11832T+1824478T21832p3T3+p6T4 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+680T+744870T2+680p3T3+p6T4 1 + 680 T + 744870 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1796T+411158T2796p3T3+p6T4 1 - 796 T + 411158 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1670T+1145410T2670p3T3+p6T4 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + 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

−8.765023127653518868449475180324, −8.552390438446042517550388443818, −8.011307591654896696318463065031, −7.928121915751284233212196480082, −7.43039046950796777063564665707, −7.28162703920366389412844994007, −6.77500121526243522454268513281, −6.10789216281293128148667186853, −5.76951821820016139012997102054, −5.29085316505930329359099282207, −4.85458081491807854820364249910, −4.75093120025812212571984131523, −3.71303461658186779984528906303, −3.50589041345024107644173498813, −3.21303841623059107067077121610, −2.55841064261338600173888301568, −1.94340258328469866153911550303, −1.41669654627431476667269737316, −0.52963839797465127935562734935, −0.35838979906258974904439614630, 0.35838979906258974904439614630, 0.52963839797465127935562734935, 1.41669654627431476667269737316, 1.94340258328469866153911550303, 2.55841064261338600173888301568, 3.21303841623059107067077121610, 3.50589041345024107644173498813, 3.71303461658186779984528906303, 4.75093120025812212571984131523, 4.85458081491807854820364249910, 5.29085316505930329359099282207, 5.76951821820016139012997102054, 6.10789216281293128148667186853, 6.77500121526243522454268513281, 7.28162703920366389412844994007, 7.43039046950796777063564665707, 7.928121915751284233212196480082, 8.011307591654896696318463065031, 8.552390438446042517550388443818, 8.765023127653518868449475180324

Graph of the ZZ-function along the critical line