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Properties

Label 4-89856-1.1-c1e2-0-14
Degree 44
Conductor 8985689856
Sign −1-1
Analytic cond. 5.729295.72929
Root an. cond. 1.547121.54712
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  − 3-s + 9-s − 13-s − 8·19-s + 2·25-s − 27-s − 8·31-s − 12·37-s + 39-s − 14·49-s + 8·57-s + 12·61-s − 4·73-s − 2·75-s + 16·79-s + 81-s + 8·93-s − 4·97-s + 8·103-s − 36·109-s + 12·111-s − 117-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.277·13-s − 1.83·19-s + 2/5·25-s − 0.192·27-s − 1.43·31-s − 1.97·37-s + 0.160·39-s − 2·49-s + 1.05·57-s + 1.53·61-s − 0.468·73-s − 0.230·75-s + 1.80·79-s + 1/9·81-s + 0.829·93-s − 0.406·97-s + 0.788·103-s − 3.44·109-s + 1.13·111-s − 0.0924·117-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(89856s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(89856s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 89856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8985689856    =    28⋅33⋅132^{8} \cdot 3^{3} \cdot 13
Sign: −1-1
Analytic conductor: 5.729295.72929
Root analytic conductor: 1.547121.54712
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 89856, ( :1/2,1/2), −1)(4,\ 89856,\ (\ :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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
3C1C_1 1+T 1 + T
13C1C_1×\timesC2C_2 (1−T)(1+2T+pT2) ( 1 - T )( 1 + 2 T + p T^{2} )
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4} 2.5.a_ac
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2} 2.7.a_o
11C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4} 2.11.a_ak
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) 2.17.a_ac
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2} 2.19.i_cc
23C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) 2.23.a_as
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4} 2.29.a_bm
31C2C_2×\timesC2C_2 (1+pT2)(1+8T+pT2) ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) 2.31.i_ck
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) 2.37.m_dq
41C22C_2^2 1−42T2+p2T4 1 - 42 T^{2} + p^{2} T^{4} 2.41.a_abq
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) 2.43.a_cs
47C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4} 2.47.a_abi
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) 2.53.a_g
59C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4} 2.59.a_ak
61C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2} 2.61.am_gc
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) 2.67.a_ak
71C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4} 2.71.a_eg
73C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) 2.73.e_fe
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2} 2.79.aq_io
83C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4} 2.83.a_aba
89C22C_2^2 1+70T2+p2T4 1 + 70 T^{2} + p^{2} T^{4} 2.89.a_cs
97C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) 2.97.e_cc
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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

−9.467496237890461451783993915929, −8.841068792972054562849056616585, −8.490614344361660813961840308516, −7.926417296360650045367588009050, −7.30393770225398842229611101295, −6.66349222619207276866844009081, −6.54196020674029332950612347695, −5.71871808820946213108219220107, −5.18282097289206777516576185904, −4.72564278819013751077310259954, −3.95597494703164348462587055696, −3.43739842273262245614290193406, −2.36904129671731002829202686194, −1.64095874171669091791461038296, 0, 1.64095874171669091791461038296, 2.36904129671731002829202686194, 3.43739842273262245614290193406, 3.95597494703164348462587055696, 4.72564278819013751077310259954, 5.18282097289206777516576185904, 5.71871808820946213108219220107, 6.54196020674029332950612347695, 6.66349222619207276866844009081, 7.30393770225398842229611101295, 7.926417296360650045367588009050, 8.490614344361660813961840308516, 8.841068792972054562849056616585, 9.467496237890461451783993915929

Graph of the ZZ-function along the critical line