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Properties

Label 8-1152e4-1.1-c3e4-0-0
Degree 88
Conductor 1.761×10121.761\times 10^{12}
Sign 11
Analytic cond. 2.13439×1072.13439\times 10^{7}
Root an. cond. 8.244408.24440
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  + 176·25-s + 1.37e3·49-s − 2.36e3·73-s − 7.26e3·97-s + 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 1.40·25-s + 4·49-s − 3.79·73-s − 7.60·97-s + 4·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

Λ(s)=((228⋅38)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((228⋅38)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅382^{28} \cdot 3^{8}
Sign: 11
Analytic conductor: 2.13439×1072.13439\times 10^{7}
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅38, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.32199584240.3219958424
L(12)L(\frac12) ≈\approx 0.32199584240.3219958424
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)
bad2 1 1
3 1 1
good5C22C_2^2 (1−88T2+p6T4)2 ( 1 - 88 T^{2} + p^{6} T^{4} )^{2}
7C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
11C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
13C2C_2 (1−92T+p3T2)2(1+92T+p3T2)2 ( 1 - 92 T + p^{3} T^{2} )^{2}( 1 + 92 T + p^{3} T^{2} )^{2}
17C22C_2^2 (1+9776T2+p6T4)2 ( 1 + 9776 T^{2} + p^{6} T^{4} )^{2}
19C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
23C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
29C22C_2^2 (1+36920T2+p6T4)2 ( 1 + 36920 T^{2} + p^{6} T^{4} )^{2}
31C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
37C2C_2 (1−214T+p3T2)2(1+214T+p3T2)2 ( 1 - 214 T + p^{3} T^{2} )^{2}( 1 + 214 T + p^{3} T^{2} )^{2}
41C22C_2^2 (1+108560T2+p6T4)2 ( 1 + 108560 T^{2} + p^{6} T^{4} )^{2}
43C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
47C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
53C22C_2^2 (1−296296T2+p6T4)2 ( 1 - 296296 T^{2} + p^{6} T^{4} )^{2}
59C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
61C2C_2 (1−830T+p3T2)2(1+830T+p3T2)2 ( 1 - 830 T + p^{3} T^{2} )^{2}( 1 + 830 T + p^{3} T^{2} )^{2}
67C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
71C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
73C2C_2 (1+592T+p3T2)4 ( 1 + 592 T + p^{3} T^{2} )^{4}
79C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
83C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
89C22C_2^2 (1−293920T2+p6T4)2 ( 1 - 293920 T^{2} + p^{6} T^{4} )^{2}
97C2C_2 (1+1816T+p3T2)4 ( 1 + 1816 T + p^{3} T^{2} )^{4}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.73422977769241169477222487143, −6.61893424048649174697038790167, −6.05287804043078955779494127528, −5.81770935679324149884641554258, −5.65507343447641795673971559459, −5.55531892634875261669235881199, −5.37008119319789691132256949802, −5.21054607338256129646492429297, −4.53555289415880912468798460296, −4.45731172314289042841743476570, −4.42964052663799843463428051341, −4.09677944643574752211361608360, −3.99793999974100759427451722994, −3.51010709223943012825312797950, −3.18134480687261045109326361615, −2.99981978123335405951308899064, −2.71305322515072008659668448732, −2.68156329337505444045101975355, −2.06498841701380841888108266480, −2.00782911250493395679079592510, −1.47445013071255324195551202670, −1.25195164224563536536488446481, −0.877705328346766892143239469982, −0.67518960414272005995134553475, −0.06517781382214689101433544798, 0.06517781382214689101433544798, 0.67518960414272005995134553475, 0.877705328346766892143239469982, 1.25195164224563536536488446481, 1.47445013071255324195551202670, 2.00782911250493395679079592510, 2.06498841701380841888108266480, 2.68156329337505444045101975355, 2.71305322515072008659668448732, 2.99981978123335405951308899064, 3.18134480687261045109326361615, 3.51010709223943012825312797950, 3.99793999974100759427451722994, 4.09677944643574752211361608360, 4.42964052663799843463428051341, 4.45731172314289042841743476570, 4.53555289415880912468798460296, 5.21054607338256129646492429297, 5.37008119319789691132256949802, 5.55531892634875261669235881199, 5.65507343447641795673971559459, 5.81770935679324149884641554258, 6.05287804043078955779494127528, 6.61893424048649174697038790167, 6.73422977769241169477222487143

Graph of the ZZ-function along the critical line