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Properties

Label 8-832e4-1.1-c3e4-0-4
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 5.80707×1065.80707\times 10^{6}
Root an. cond. 7.006397.00639
Motivic weight 33
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  + 100·9-s + 456·17-s + 436·25-s − 1.22e3·49-s + 6.04e3·81-s + 7.27e3·113-s − 5.32e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4.56e4·153-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 3.70·9-s + 6.50·17-s + 3.48·25-s − 3.58·49-s + 8.28·81-s + 6.05·113-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 + 24.0·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-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 + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 5.80707×1065.80707\times 10^{6}
Root analytic conductor: 7.006397.00639
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 24.9505192024.95051920
L(12)L(\frac12) ≈\approx 24.9505192024.95051920
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
13C2C_2 (1−p3T2)2 ( 1 - p^{3} T^{2} )^{2}
good3C22C_2^2 (1−50T2+p6T4)2 ( 1 - 50 T^{2} + p^{6} T^{4} )^{2}
5C22C_2^2 (1−218T2+p6T4)2 ( 1 - 218 T^{2} + p^{6} T^{4} )^{2}
7C22C_2^2 (1+614T2+p6T4)2 ( 1 + 614 T^{2} + p^{6} T^{4} )^{2}
11C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
17C2C_2 (1−114T+p3T2)4 ( 1 - 114 T + p^{3} T^{2} )^{4}
19C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
23C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
29C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
31C22C_2^2 (1+27830T2+p6T4)2 ( 1 + 27830 T^{2} + p^{6} T^{4} )^{2}
37C22C_2^2 (1+13894T2+p6T4)2 ( 1 + 13894 T^{2} + p^{6} T^{4} )^{2}
41C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
43C22C_2^2 (1−111490T2+p6T4)2 ( 1 - 111490 T^{2} + p^{6} T^{4} )^{2}
47C22C_2^2 (1−128554T2+p6T4)2 ( 1 - 128554 T^{2} + p^{6} T^{4} )^{2}
53C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
59C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
61C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
67C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
71C22C_2^2 (1+317990T2+p6T4)2 ( 1 + 317990 T^{2} + p^{6} T^{4} )^{2}
73C2C_2 (1−p3T2)4 ( 1 - 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}
89C2C_2 (1−p3T2)4 ( 1 - p^{3} T^{2} )^{4}
97C2C_2 (1−p3T2)4 ( 1 - 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

−7.14646700667869170074277343000, −6.77035291564498937452429560165, −6.35664858476086211217166763430, −6.29632250044155556394084103502, −6.24335216661523665124391168182, −5.60739322719269310068746873419, −5.34495591419483886972721416974, −5.21904066203221727610461113217, −5.20263459411241994324494870842, −4.72614439732522367276454308747, −4.58447641022976690460712429427, −4.41179300902422430406136931806, −3.96045027542235399739548356545, −3.68115063138233718562954566666, −3.40514440466808035128478991125, −3.24968231482626736720406780360, −3.03470568824294042616519003515, −2.97981169358095153220363573530, −2.15539189522419466948131426337, −1.81199409212303256734610448942, −1.45257892079413220735776525317, −1.20179157497760076143325757524, −1.13411660648558669153442893813, −0.905278196903914937964373120889, −0.55864329336077472517644823545, 0.55864329336077472517644823545, 0.905278196903914937964373120889, 1.13411660648558669153442893813, 1.20179157497760076143325757524, 1.45257892079413220735776525317, 1.81199409212303256734610448942, 2.15539189522419466948131426337, 2.97981169358095153220363573530, 3.03470568824294042616519003515, 3.24968231482626736720406780360, 3.40514440466808035128478991125, 3.68115063138233718562954566666, 3.96045027542235399739548356545, 4.41179300902422430406136931806, 4.58447641022976690460712429427, 4.72614439732522367276454308747, 5.20263459411241994324494870842, 5.21904066203221727610461113217, 5.34495591419483886972721416974, 5.60739322719269310068746873419, 6.24335216661523665124391168182, 6.29632250044155556394084103502, 6.35664858476086211217166763430, 6.77035291564498937452429560165, 7.14646700667869170074277343000

Graph of the ZZ-function along the critical line