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Properties

Label 4-2352e2-1.1-c1e2-0-1
Degree 44
Conductor 55319045531904
Sign 11
Analytic cond. 352.718352.718
Root an. cond. 4.333684.33368
Motivic weight 11
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  − 3-s − 6·5-s + 6·11-s + 6·15-s + 5·19-s − 12·23-s + 19·25-s + 27-s − 5·31-s − 6·33-s + 11·37-s − 6·47-s − 12·53-s − 36·55-s − 5·57-s + 12·59-s + 24·61-s − 15·67-s + 12·69-s + 9·73-s − 19·75-s − 21·79-s − 81-s − 36·83-s − 12·89-s + 5·93-s − 30·95-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.68·5-s + 1.80·11-s + 1.54·15-s + 1.14·19-s − 2.50·23-s + 19/5·25-s + 0.192·27-s − 0.898·31-s − 1.04·33-s + 1.80·37-s − 0.875·47-s − 1.64·53-s − 4.85·55-s − 0.662·57-s + 1.56·59-s + 3.07·61-s − 1.83·67-s + 1.44·69-s + 1.05·73-s − 2.19·75-s − 2.36·79-s − 1/9·81-s − 3.95·83-s − 1.27·89-s + 0.518·93-s − 3.07·95-s + ⋯

Functional equation

Λ(s)=(5531904s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5531904s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 55319045531904    =    28⋅32⋅742^{8} \cdot 3^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 352.718352.718
Root analytic conductor: 4.333684.33368
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 5531904, ( :1/2,1/2), 1)(4,\ 5531904,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.29972629320.2997262932
L(12)L(\frac12) ≈\approx 0.29972629320.2997262932
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)
bad2 1 1
3C2C_2 1+T+T2 1 + T + T^{2}
7 1 1
good5C22C_2^2 1+6T+17T2+6pT3+p2T4 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
13C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
19C22C_2^2 1−5T+6T2−5pT3+p2T4 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+12T+71T2+12pT3+p2T4 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4}
29C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
31C22C_2^2 1+5T−6T2+5pT3+p2T4 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4}
37C2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
41C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
43C22C_2^2 1−11T2+p2T4 1 - 11 T^{2} + p^{2} T^{4}
47C22C_2^2 1+6T−11T2+6pT3+p2T4 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+12T+91T2+12pT3+p2T4 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−12T+85T2−12pT3+p2T4 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−24T+253T2−24pT3+p2T4 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+15T+142T2+15pT3+p2T4 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
73C22C_2^2 1−9T+100T2−9pT3+p2T4 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4}
79C2C_2 (1+4T+pT2)(1+17T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C2C_2 (1+18T+pT2)2 ( 1 + 18 T + p T^{2} )^{2}
89C22C_2^2 1+12T+137T2+12pT3+p2T4 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−146T2+p2T4 1 - 146 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

−9.671698580830243451694689247146, −8.528595555660986011144992333201, −8.449942892896759581205198928803, −7.88629172481279021040580104344, −7.78290243696478170870578388742, −7.17041631603365486434514503538, −7.06711350784399495827442739540, −6.48775595233884848843344908526, −6.20002311406581522914434435362, −5.47641920194200923920322113295, −5.42210625503084125942408110976, −4.36821327415526520552638126455, −4.35880064904406542696236399975, −3.84715448270943117777029326617, −3.84558346434773473541643891978, −3.12720229769520223265923479699, −2.63638186442779382525477860321, −1.59631233487813131159050231603, −1.12382345083544906122864893634, −0.22836289856137411672963545183, 0.22836289856137411672963545183, 1.12382345083544906122864893634, 1.59631233487813131159050231603, 2.63638186442779382525477860321, 3.12720229769520223265923479699, 3.84558346434773473541643891978, 3.84715448270943117777029326617, 4.35880064904406542696236399975, 4.36821327415526520552638126455, 5.42210625503084125942408110976, 5.47641920194200923920322113295, 6.20002311406581522914434435362, 6.48775595233884848843344908526, 7.06711350784399495827442739540, 7.17041631603365486434514503538, 7.78290243696478170870578388742, 7.88629172481279021040580104344, 8.449942892896759581205198928803, 8.528595555660986011144992333201, 9.671698580830243451694689247146

Graph of the ZZ-function along the critical line