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Properties

Label 8-3267e4-1.1-c0e4-0-3
Degree 88
Conductor 1.139×10141.139\times 10^{14}
Sign 11
Analytic cond. 7.066837.06683
Root an. cond. 1.276881.27688
Motivic weight 00
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  + 4·4-s + 10·16-s − 4·25-s + 20·64-s + 4·97-s − 16·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 4·4-s + 10·16-s − 4·25-s + 20·64-s + 4·97-s − 16·100-s + 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯

Functional equation

Λ(s)=((312⋅118)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((312⋅118)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 312⋅1183^{12} \cdot 11^{8}
Sign: 11
Analytic conductor: 7.066837.06683
Root analytic conductor: 1.276881.27688
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 312⋅118, ( :0,0,0,0), 1)(8,\ 3^{12} \cdot 11^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 6.0583720856.058372085
L(12)L(\frac12) ≈\approx 6.0583720856.058372085
L(1)L(1) 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)
bad3 1 1
11 1 1
good2C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
5C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
7C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
13C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
17C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
19C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
23C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
29C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
31C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
37C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
41C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
43C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
47C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
53C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
59C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
61C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
67C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
71C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
73C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
79C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
83C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
89C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
97C2C_2 (1−T+T2)4 ( 1 - T + 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

−6.47896242437598672541613328868, −6.10000461034407803781073997851, −5.96527130959385211439344471664, −5.93224957270222222737158279360, −5.71557868154761541621362927897, −5.38588173126502152311447486104, −5.29381553457790105101882656502, −5.05556988636564458177397510377, −4.78494384024374958637999385871, −4.41526534156070664886655197547, −4.10294859997713759860091138422, −3.83425013407645239092474736472, −3.82070752023145197271420709173, −3.48356403681026982115537413227, −3.29401099533779911087651522055, −3.05550009227584157588296114351, −2.94873373982252916440742829324, −2.58563405145722755519026087337, −2.22634970037347322430525235748, −2.08904261085142173650190533901, −1.99624627511606155583601100987, −1.80857182052585382954070619772, −1.62842713776413164374217909733, −0.987091595782885604660525575453, −0.895633476524245239422701637981, 0.895633476524245239422701637981, 0.987091595782885604660525575453, 1.62842713776413164374217909733, 1.80857182052585382954070619772, 1.99624627511606155583601100987, 2.08904261085142173650190533901, 2.22634970037347322430525235748, 2.58563405145722755519026087337, 2.94873373982252916440742829324, 3.05550009227584157588296114351, 3.29401099533779911087651522055, 3.48356403681026982115537413227, 3.82070752023145197271420709173, 3.83425013407645239092474736472, 4.10294859997713759860091138422, 4.41526534156070664886655197547, 4.78494384024374958637999385871, 5.05556988636564458177397510377, 5.29381553457790105101882656502, 5.38588173126502152311447486104, 5.71557868154761541621362927897, 5.93224957270222222737158279360, 5.96527130959385211439344471664, 6.10000461034407803781073997851, 6.47896242437598672541613328868

Graph of the ZZ-function along the critical line