Properties

Label 4-1323e2-1.1-c3e2-0-2
Degree 44
Conductor 17503291750329
Sign 11
Analytic cond. 6093.286093.28
Root an. cond. 8.835138.83513
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  − 6·2-s + 14·4-s + 6·5-s − 36·10-s − 48·11-s − 52·13-s − 84·16-s + 30·17-s − 64·19-s + 84·20-s + 288·22-s − 60·23-s − 175·25-s + 312·26-s − 360·29-s + 140·31-s + 216·32-s − 180·34-s − 230·37-s + 384·38-s + 234·41-s − 938·43-s − 672·44-s + 360·46-s + 618·47-s + 1.05e3·50-s − 728·52-s + ⋯
L(s)  = 1  − 2.12·2-s + 7/4·4-s + 0.536·5-s − 1.13·10-s − 1.31·11-s − 1.10·13-s − 1.31·16-s + 0.428·17-s − 0.772·19-s + 0.939·20-s + 2.79·22-s − 0.543·23-s − 7/5·25-s + 2.35·26-s − 2.30·29-s + 0.811·31-s + 1.19·32-s − 0.907·34-s − 1.02·37-s + 1.63·38-s + 0.891·41-s − 3.32·43-s − 2.30·44-s + 1.15·46-s + 1.91·47-s + 2.96·50-s − 1.94·52-s + ⋯

Functional equation

Λ(s)=(1750329s/2ΓC(s)2L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1750329s/2ΓC(s+3/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 17503291750329    =    36743^{6} \cdot 7^{4}
Sign: 11
Analytic conductor: 6093.286093.28
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1750329, ( :3/2,3/2), 1)(4,\ 1750329,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) \approx 0.32804864920.3280486492
L(12)L(\frac12) \approx 0.32804864920.3280486492
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
7 1 1
good2D4D_{4} 1+3pT+11pT2+3p4T3+p6T4 1 + 3 p T + 11 p T^{2} + 3 p^{4} T^{3} + p^{6} T^{4}
5D4D_{4} 16T+211T26p3T3+p6T4 1 - 6 T + 211 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+48T+2470T2+48p3T3+p6T4 1 + 48 T + 2470 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+4pT+3342T2+4p4T3+p6T4 1 + 4 p T + 3342 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 130T+7699T230p3T3+p6T4 1 - 30 T + 7699 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+64T+3942T2+64p3T3+p6T4 1 + 64 T + 3942 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+60T+494pT2+60p3T3+p6T4 1 + 60 T + 494 p T^{2} + 60 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+360T+77290T2+360p3T3+p6T4 1 + 360 T + 77290 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1140T+48930T2140p3T3+p6T4 1 - 140 T + 48930 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+230T+98979T2+230p3T3+p6T4 1 + 230 T + 98979 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1234T+26683T2234p3T3+p6T4 1 - 234 T + 26683 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+938T+378543T2+938p3T3+p6T4 1 + 938 T + 378543 T^{2} + 938 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1618T+210199T2618p3T3+p6T4 1 - 618 T + 210199 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1420T+64606T2420p3T3+p6T4 1 - 420 T + 64606 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1282T+411439T2282p3T3+p6T4 1 - 282 T + 411439 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 132T+205386T232p3T3+p6T4 1 - 32 T + 205386 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1544T+535542T2544p3T3+p6T4 1 - 544 T + 535542 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1504T+736126T2504p3T3+p6T4 1 - 504 T + 736126 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1764T+653958T2764p3T3+p6T4 1 - 764 T + 653958 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1238T+125439T2238p3T3+p6T4 1 - 238 T + 125439 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+522T+651823T2+522p3T3+p6T4 1 + 522 T + 651823 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1708T+1163542T2708p3T3+p6T4 1 - 708 T + 1163542 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+664T+1779618T2+664p3T3+p6T4 1 + 664 T + 1779618 T^{2} + 664 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

−9.454797620452498735747298468023, −9.201469306942394824899384266169, −8.656354916581939464317960395147, −8.255240893443178631128607904750, −7.908415229351685857426441268681, −7.78191408827438268485925146769, −7.23983179163634404191904145114, −6.85153724283759710360390840084, −6.37468083428123473232078564898, −5.61261705391939498576129662142, −5.31722524763747337099803341728, −5.15649706773807242275127136286, −4.14894315093212786988719565516, −3.97084032343759934370877409845, −3.11144669706821286879265207432, −2.42981308880365942153631857327, −1.85913980337783281874636711339, −1.81117132762125842445054730865, −0.61833833769799951755914455817, −0.31136106903687004700543830908, 0.31136106903687004700543830908, 0.61833833769799951755914455817, 1.81117132762125842445054730865, 1.85913980337783281874636711339, 2.42981308880365942153631857327, 3.11144669706821286879265207432, 3.97084032343759934370877409845, 4.14894315093212786988719565516, 5.15649706773807242275127136286, 5.31722524763747337099803341728, 5.61261705391939498576129662142, 6.37468083428123473232078564898, 6.85153724283759710360390840084, 7.23983179163634404191904145114, 7.78191408827438268485925146769, 7.908415229351685857426441268681, 8.255240893443178631128607904750, 8.656354916581939464317960395147, 9.201469306942394824899384266169, 9.454797620452498735747298468023

Graph of the ZZ-function along the critical line