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Properties

Label 4-170e2-1.1-c1e2-0-12
Degree 44
Conductor 2890028900
Sign −1-1
Analytic cond. 1.842681.84268
Root an. cond. 1.165091.16509
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s − 4·13-s − 16-s − 6·17-s − 8·19-s − 25-s + 4·26-s − 5·32-s + 6·34-s + 8·38-s + 2·43-s + 18·47-s − 8·49-s + 50-s + 4·52-s − 8·53-s − 16·59-s + 7·64-s + 6·67-s + 6·68-s + 8·76-s − 9·81-s − 26·83-s − 2·86-s − 18·94-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.10·13-s − 1/4·16-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.784·26-s − 0.883·32-s + 1.02·34-s + 1.29·38-s + 0.304·43-s + 2.62·47-s − 8/7·49-s + 0.141·50-s + 0.554·52-s − 1.09·53-s − 2.08·59-s + 7/8·64-s + 0.733·67-s + 0.727·68-s + 0.917·76-s − 81-s − 2.85·83-s − 0.215·86-s − 1.85·94-s + ⋯

Functional equation

Λ(s)=(28900s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(28900s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2890028900    =    22⋅52⋅1722^{2} \cdot 5^{2} \cdot 17^{2}
Sign: −1-1
Analytic conductor: 1.842681.84268
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 28900, ( :1/2,1/2), −1)(4,\ 28900,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad2C2C_2 1+T+pT2 1 + T + p T^{2}
5C2C_2 1+T2 1 + T^{2}
17C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good3C22C_2^2 1+p2T4 1 + p^{2} T^{4}
7C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
11C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} )
23C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
29C22C_2^2 1+30T2+p2T4 1 + 30 T^{2} + p^{2} T^{4}
31C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
37C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
41C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−12T+pT2)(1+10T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} )
47C2C_2×\timesC2C_2 (1−10T+pT2)(1−8T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} )
53C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2×\timesC2C_2 (1+4T+pT2)(1+12T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−10T+pT2)(1+4T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C22C_2^2 1−72T2+p2T4 1 - 72 T^{2} + p^{2} T^{4}
73C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
79C22C_2^2 1+120T2+p2T4 1 + 120 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1+12T+pT2)(1+14T+pT2) ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} )
89C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
97C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.21297280232623962707754863672, −9.708254981814360858292663699128, −9.202072821355533222558179266565, −8.662434285870896733895612797810, −8.423005787233979230346840942154, −7.53639490843311349121052775111, −7.27812164456876842814157434085, −6.46471330296411122639951342724, −5.93303168624518243897643367682, −4.94139277905990953926192582146, −4.48404698138439520044486747643, −3.99612885944062491515261866857, −2.68559110089798692482543213228, −1.85990320101749540748192083329, 0, 1.85990320101749540748192083329, 2.68559110089798692482543213228, 3.99612885944062491515261866857, 4.48404698138439520044486747643, 4.94139277905990953926192582146, 5.93303168624518243897643367682, 6.46471330296411122639951342724, 7.27812164456876842814157434085, 7.53639490843311349121052775111, 8.423005787233979230346840942154, 8.662434285870896733895612797810, 9.202072821355533222558179266565, 9.708254981814360858292663699128, 10.21297280232623962707754863672

Graph of the ZZ-function along the critical line