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Properties

Label 4-9360e2-1.1-c1e2-0-17
Degree 44
Conductor 8760960087609600
Sign 11
Analytic cond. 5586.065586.06
Root an. cond. 8.645228.64522
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s − 2·13-s + 4·17-s − 4·19-s − 4·23-s + 3·25-s − 4·29-s − 8·35-s + 4·37-s − 4·41-s − 8·47-s − 2·49-s − 8·59-s − 4·61-s + 4·65-s + 16·67-s − 16·73-s + 16·79-s − 16·83-s − 8·85-s − 4·89-s − 8·91-s + 8·95-s − 28·101-s + 16·103-s − 16·107-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 3/5·25-s − 0.742·29-s − 1.35·35-s + 0.657·37-s − 0.624·41-s − 1.16·47-s − 2/7·49-s − 1.04·59-s − 0.512·61-s + 0.496·65-s + 1.95·67-s − 1.87·73-s + 1.80·79-s − 1.75·83-s − 0.867·85-s − 0.423·89-s − 0.838·91-s + 0.820·95-s − 2.78·101-s + 1.57·103-s − 1.54·107-s + ⋯

Functional equation

Λ(s)=(87609600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(87609600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8760960087609600    =    28⋅34⋅52⋅1322^{8} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 5586.065586.06
Root analytic conductor: 8.645228.64522
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 87609600, ( :1/2,1/2), 1)(4,\ 87609600,\ (\ :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)
bad2 1 1
3 1 1
5C1C_1 (1+T)2 ( 1 + T )^{2}
13C1C_1 (1+T)2 ( 1 + T )^{2}
good7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17D4D_{4} 1−4T+18T2−4pT3+p2T4 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23D4D_{4} 1+4T+30T2+4pT3+p2T4 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+4T+42T2+4pT3+p2T4 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+42T2+p2T4 1 + 42 T^{2} + p^{2} T^{4}
37D4D_{4} 1−4T−2T2−4pT3+p2T4 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4}
41C4C_4 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
43C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
47C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
53C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
59D4D_{4} 1+8T+54T2+8pT3+p2T4 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4}
61D4D_{4} 1+4T+46T2+4pT3+p2T4 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−16T+178T2−16pT3+p2T4 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73D4D_{4} 1+16T+190T2+16pT3+p2T4 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−16T+142T2−16pT3+p2T4 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4}
83C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C22C_2^2 1+174T2+p2T4 1 + 174 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

−7.53928565741736582862105781192, −7.36760705411490076301331171588, −6.90763472019682130761682656765, −6.60909506464236844769008738881, −6.11732228827021399748291447723, −5.85721328381193558326302688943, −5.24335244464295165360164816652, −5.20940502283961275252602590700, −4.70302288921122313140494575231, −4.39109677054041065407648951287, −4.13868200220963801418778627032, −3.69289030780213072854720058025, −3.24656157079255885764273954366, −2.93340971749375441862981186755, −2.22369602642011174801552817016, −2.04776720337917005438234830180, −1.30646498703490264752802920334, −1.23372236321440538539139824725, 0, 0, 1.23372236321440538539139824725, 1.30646498703490264752802920334, 2.04776720337917005438234830180, 2.22369602642011174801552817016, 2.93340971749375441862981186755, 3.24656157079255885764273954366, 3.69289030780213072854720058025, 4.13868200220963801418778627032, 4.39109677054041065407648951287, 4.70302288921122313140494575231, 5.20940502283961275252602590700, 5.24335244464295165360164816652, 5.85721328381193558326302688943, 6.11732228827021399748291447723, 6.60909506464236844769008738881, 6.90763472019682130761682656765, 7.36760705411490076301331171588, 7.53928565741736582862105781192

Graph of the ZZ-function along the critical line