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Properties

Label 8-832e4-1.1-c2e4-0-0
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 264139.264139.
Root an. cond. 4.761334.76133
Motivic weight 22
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  − 10·9-s + 40·13-s − 20·17-s − 38·25-s − 160·29-s − 46·49-s + 40·53-s + 160·61-s − 87·81-s + 208·101-s + 40·113-s − 400·117-s − 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 200·153-s + 157-s + 163-s + 167-s + 862·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.11·9-s + 3.07·13-s − 1.17·17-s − 1.51·25-s − 5.51·29-s − 0.938·49-s + 0.754·53-s + 2.62·61-s − 1.07·81-s + 2.05·101-s + 0.353·113-s − 3.41·117-s − 0.826·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.30·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 5.10·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

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

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 264139.264139.
Root analytic conductor: 4.761334.76133
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1,1,1,1), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.2675470262.267547026
L(12)L(\frac12) ≈\approx 2.2675470262.267547026
L(2)L(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 (1−20T+p2T2)2 ( 1 - 20 T + p^{2} T^{2} )^{2}
good3C22C_2^2 (1+5T2+p4T4)2 ( 1 + 5 T^{2} + p^{4} T^{4} )^{2}
5C22C_2^2 (1+19T2+p4T4)2 ( 1 + 19 T^{2} + p^{4} T^{4} )^{2}
7C22C_2^2 (1+23T2+p4T4)2 ( 1 + 23 T^{2} + p^{4} T^{4} )^{2}
11C22C_2^2 (1+50T2+p4T4)2 ( 1 + 50 T^{2} + p^{4} T^{4} )^{2}
17C2C_2 (1+5T+p2T2)4 ( 1 + 5 T + p^{2} T^{2} )^{4}
19C22C_2^2 (1+710T2+p4T4)2 ( 1 + 710 T^{2} + p^{4} T^{4} )^{2}
23C22C_2^2 (1−10pT2+p4T4)2 ( 1 - 10 p T^{2} + p^{4} T^{4} )^{2}
29C2C_2 (1+40T+p2T2)4 ( 1 + 40 T + p^{2} T^{2} )^{4}
31C22C_2^2 (1+1490T2+p4T4)2 ( 1 + 1490 T^{2} + p^{4} T^{4} )^{2}
37C22C_2^2 (1−1013T2+p4T4)2 ( 1 - 1013 T^{2} + p^{4} T^{4} )^{2}
41C1C_1×\timesC1C_1 (1−pT)4(1+pT)4 ( 1 - p T )^{4}( 1 + p T )^{4}
43C22C_2^2 (1−1835T2+p4T4)2 ( 1 - 1835 T^{2} + p^{4} T^{4} )^{2}
47C22C_2^2 (1−1657T2+p4T4)2 ( 1 - 1657 T^{2} + p^{4} T^{4} )^{2}
53C2C_2 (1−10T+p2T2)4 ( 1 - 10 T + p^{2} T^{2} )^{4}
59C22C_2^2 (1+2630T2+p4T4)2 ( 1 + 2630 T^{2} + p^{4} T^{4} )^{2}
61C2C_2 (1−40T+p2T2)4 ( 1 - 40 T + p^{2} T^{2} )^{4}
67C22C_2^2 (1−1822T2+p4T4)2 ( 1 - 1822 T^{2} + p^{4} T^{4} )^{2}
71C22C_2^2 (1+9215T2+p4T4)2 ( 1 + 9215 T^{2} + p^{4} T^{4} )^{2}
73C22C_2^2 (1−3758T2+p4T4)2 ( 1 - 3758 T^{2} + p^{4} T^{4} )^{2}
79C22C_2^2 (1+8218T2+p4T4)2 ( 1 + 8218 T^{2} + p^{4} T^{4} )^{2}
83C22C_2^2 (1+13478T2+p4T4)2 ( 1 + 13478 T^{2} + p^{4} T^{4} )^{2}
89C22C_2^2 (1−8942T2+p4T4)2 ( 1 - 8942 T^{2} + p^{4} T^{4} )^{2}
97C22C_2^2 (1−8882T2+p4T4)2 ( 1 - 8882 T^{2} + p^{4} T^{4} )^{2}
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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

−7.12602520888559376802844701154, −6.88475691733814945617030800399, −6.63623073584368899095370878068, −6.44232578129918051875450237927, −5.95172124297524132111167581998, −5.88645589491455845466279872257, −5.79210771237298611931130248238, −5.74981641944274741796422330244, −5.29858153473202286045862479321, −5.17175397934008883980612242611, −4.82506902574281737094248580334, −4.24554752547903162917047911807, −4.05926862552703457064648489796, −4.00401524844037044231925040550, −3.72709319612568352332784464540, −3.43735675201342973265369876563, −3.32519894900346866695385289130, −2.99075512472621930911330746437, −2.25771636584362839126097995097, −2.23287153949932623455361191369, −1.94118205391733956502527554885, −1.52227262690839896806596496050, −1.33385465951921127242116032953, −0.46199678033481057364057735260, −0.37264892199664266296029216095, 0.37264892199664266296029216095, 0.46199678033481057364057735260, 1.33385465951921127242116032953, 1.52227262690839896806596496050, 1.94118205391733956502527554885, 2.23287153949932623455361191369, 2.25771636584362839126097995097, 2.99075512472621930911330746437, 3.32519894900346866695385289130, 3.43735675201342973265369876563, 3.72709319612568352332784464540, 4.00401524844037044231925040550, 4.05926862552703457064648489796, 4.24554752547903162917047911807, 4.82506902574281737094248580334, 5.17175397934008883980612242611, 5.29858153473202286045862479321, 5.74981641944274741796422330244, 5.79210771237298611931130248238, 5.88645589491455845466279872257, 5.95172124297524132111167581998, 6.44232578129918051875450237927, 6.63623073584368899095370878068, 6.88475691733814945617030800399, 7.12602520888559376802844701154

Graph of the ZZ-function along the critical line