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Properties

Label 8-384e4-1.1-c3e4-0-3
Degree 88
Conductor 2174327193621743271936
Sign 11
Analytic cond. 263505.263505.
Root an. cond. 4.759904.75990
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  − 46·9-s − 500·25-s + 1.37e3·49-s + 1.72e3·73-s + 1.38e3·81-s − 7.64e3·97-s + 4.67e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2.30e4·225-s + ⋯
L(s)  = 1  − 1.70·9-s − 4·25-s + 4·49-s + 2.75·73-s + 1.90·81-s − 7.99·97-s + 3.51·121-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 + 4·169-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 + 6.81·225-s + ⋯

Functional equation

Λ(s)=((228⋅34)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((228⋅34)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅342^{28} \cdot 3^{4}
Sign: 11
Analytic conductor: 263505.263505.
Root analytic conductor: 4.759904.75990
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.77530724510.7753072451
L(12)L(\frac12) ≈\approx 0.77530724510.7753072451
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
3C22C_2^2 1+46T2+p6T4 1 + 46 T^{2} + p^{6} T^{4}
good5C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
7C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
11C22C_2^2 (1−2338T2+p6T4)2 ( 1 - 2338 T^{2} + p^{6} T^{4} )^{2}
13C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
17C2C_2 (1−90T+p3T2)2(1+90T+p3T2)2 ( 1 - 90 T + p^{3} T^{2} )^{2}( 1 + 90 T + p^{3} T^{2} )^{2}
19C22C_2^2 (1−2482T2+p6T4)2 ( 1 - 2482 T^{2} + p^{6} T^{4} )^{2}
23C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
29C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
31C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
37C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
41C2C_2 (1−522T+p3T2)2(1+522T+p3T2)2 ( 1 - 522 T + p^{3} T^{2} )^{2}( 1 + 522 T + p^{3} T^{2} )^{2}
43C22C_2^2 (1−74914T2+p6T4)2 ( 1 - 74914 T^{2} + p^{6} T^{4} )^{2}
47C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
53C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
59C22C_2^2 (1+304958T2+p6T4)2 ( 1 + 304958 T^{2} + p^{6} T^{4} )^{2}
61C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
67C22C_2^2 (1−596626T2+p6T4)2 ( 1 - 596626 T^{2} + p^{6} T^{4} )^{2}
71C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
73C2C_2 (1−430T+p3T2)4 ( 1 - 430 T + p^{3} T^{2} )^{4}
79C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
83C22C_2^2 (1+678926T2+p6T4)2 ( 1 + 678926 T^{2} + p^{6} T^{4} )^{2}
89C2C_2 (1−1026T+p3T2)2(1+1026T+p3T2)2 ( 1 - 1026 T + p^{3} T^{2} )^{2}( 1 + 1026 T + p^{3} T^{2} )^{2}
97C2C_2 (1+1910T+p3T2)4 ( 1 + 1910 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

−7.933727857767893288667867694332, −7.50513498166158157610022203528, −7.28865293552701875292825605761, −7.09107231736497162640078807037, −6.77005270257119978712905172653, −6.48448745771862606971275288422, −6.16634286078935731291759537324, −5.77824610563574928417970298036, −5.72773760055005578529838852379, −5.58079139905286779157066356359, −5.36909914044477821358717535262, −5.06102458415549858583442647704, −4.39714878560795207867375465301, −4.32560239626575882758833640778, −3.87032060020497671754576868261, −3.84021682710577073892891907720, −3.51666678436270297953652596006, −2.90019593020829020969599279552, −2.77506563244789333450532936977, −2.35459173960219017974957033223, −2.05384414727084546337938854870, −1.79593205692884855498555751140, −1.12219058602106780854641354648, −0.61095474250302536327077017672, −0.18303542954187130978219015007, 0.18303542954187130978219015007, 0.61095474250302536327077017672, 1.12219058602106780854641354648, 1.79593205692884855498555751140, 2.05384414727084546337938854870, 2.35459173960219017974957033223, 2.77506563244789333450532936977, 2.90019593020829020969599279552, 3.51666678436270297953652596006, 3.84021682710577073892891907720, 3.87032060020497671754576868261, 4.32560239626575882758833640778, 4.39714878560795207867375465301, 5.06102458415549858583442647704, 5.36909914044477821358717535262, 5.58079139905286779157066356359, 5.72773760055005578529838852379, 5.77824610563574928417970298036, 6.16634286078935731291759537324, 6.48448745771862606971275288422, 6.77005270257119978712905172653, 7.09107231736497162640078807037, 7.28865293552701875292825605761, 7.50513498166158157610022203528, 7.933727857767893288667867694332

Graph of the ZZ-function along the critical line