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Properties

Label 4-1200e2-1.1-c2e2-0-11
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 1069.131069.13
Root an. cond. 5.718185.71818
Motivic weight 22
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  + 2·3-s + 14·7-s − 5·9-s − 50·13-s + 14·19-s + 28·21-s − 28·27-s + 14·31-s + 4·37-s − 100·39-s − 82·43-s + 49·49-s + 28·57-s − 2·61-s − 70·63-s − 34·67-s − 140·73-s + 116·79-s − 11·81-s − 700·91-s + 28·93-s − 98·97-s + 308·103-s − 50·109-s + 8·111-s + 250·117-s + 170·121-s + ⋯
L(s)  = 1  + 2/3·3-s + 2·7-s − 5/9·9-s − 3.84·13-s + 0.736·19-s + 4/3·21-s − 1.03·27-s + 0.451·31-s + 4/37·37-s − 2.56·39-s − 1.90·43-s + 49-s + 0.491·57-s − 0.0327·61-s − 1.11·63-s − 0.507·67-s − 1.91·73-s + 1.46·79-s − 0.135·81-s − 7.69·91-s + 0.301·93-s − 1.01·97-s + 2.99·103-s − 0.458·109-s + 0.0720·111-s + 2.13·117-s + 1.40·121-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 1069.131069.13
Root analytic conductor: 5.718185.71818
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1440000, ( :1,1), 1)(4,\ 1440000,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.7456354251.745635425
L(12)L(\frac12) ≈\approx 1.7456354251.745635425
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
3C2C_2 1−2T+p2T2 1 - 2 T + p^{2} T^{2}
5 1 1
good7C2C_2 (1−pT+p2T2)2 ( 1 - p T + p^{2} T^{2} )^{2}
11C22C_2^2 1−170T2+p4T4 1 - 170 T^{2} + p^{4} T^{4}
13C2C_2 (1+25T+p2T2)2 ( 1 + 25 T + p^{2} T^{2} )^{2}
17C22C_2^2 1+70T2+p4T4 1 + 70 T^{2} + p^{4} T^{4}
19C2C_2 (1−7T+p2T2)2 ( 1 - 7 T + p^{2} T^{2} )^{2}
23C22C_2^2 1−410T2+p4T4 1 - 410 T^{2} + p^{4} T^{4}
29C22C_2^2 1+118T2+p4T4 1 + 118 T^{2} + p^{4} T^{4}
31C2C_2 (1−7T+p2T2)2 ( 1 - 7 T + p^{2} T^{2} )^{2}
37C2C_2 (1−2T+p2T2)2 ( 1 - 2 T + p^{2} T^{2} )^{2}
41C22C_2^2 1−3290T2+p4T4 1 - 3290 T^{2} + p^{4} T^{4}
43C2C_2 (1+41T+p2T2)2 ( 1 + 41 T + p^{2} T^{2} )^{2}
47C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
53C22C_2^2 1−2090T2+p4T4 1 - 2090 T^{2} + p^{4} T^{4}
59C22C_2^2 1−5810T2+p4T4 1 - 5810 T^{2} + p^{4} T^{4}
61C2C_2 (1+T+p2T2)2 ( 1 + T + p^{2} T^{2} )^{2}
67C2C_2 (1+17T+p2T2)2 ( 1 + 17 T + p^{2} T^{2} )^{2}
71C22C_2^2 1−8282T2+p4T4 1 - 8282 T^{2} + p^{4} T^{4}
73C2C_2 (1+70T+p2T2)2 ( 1 + 70 T + p^{2} T^{2} )^{2}
79C2C_2 (1−58T+p2T2)2 ( 1 - 58 T + p^{2} T^{2} )^{2}
83C22C_2^2 1+334T2+p4T4 1 + 334 T^{2} + p^{4} T^{4}
89C22C_2^2 1+2590T2+p4T4 1 + 2590 T^{2} + p^{4} T^{4}
97C2C_2 (1+49T+p2T2)2 ( 1 + 49 T + p^{2} T^{2} )^{2}
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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

−9.654369544641867486065104388231, −9.555105942634100894442716820632, −8.757383274047098096649485132603, −8.594702413198485034848673343216, −7.902510608314406486550306029786, −7.86690159070136632162818382135, −7.42847093444129848156348725171, −7.12138683347527511915717219109, −6.62009533280546283297034795974, −5.74585019278562682775698971892, −5.42038965846699785731010433713, −4.86385638206789412012123959107, −4.71447875438696003659017012855, −4.46878811052058189696630149089, −3.38616780169582751190589176500, −3.02377698141610002159515607062, −2.29869132523221975749581522045, −2.14737934543105428712629867665, −1.46759905849771493921514604054, −0.35354229502205263785839925885, 0.35354229502205263785839925885, 1.46759905849771493921514604054, 2.14737934543105428712629867665, 2.29869132523221975749581522045, 3.02377698141610002159515607062, 3.38616780169582751190589176500, 4.46878811052058189696630149089, 4.71447875438696003659017012855, 4.86385638206789412012123959107, 5.42038965846699785731010433713, 5.74585019278562682775698971892, 6.62009533280546283297034795974, 7.12138683347527511915717219109, 7.42847093444129848156348725171, 7.86690159070136632162818382135, 7.902510608314406486550306029786, 8.594702413198485034848673343216, 8.757383274047098096649485132603, 9.555105942634100894442716820632, 9.654369544641867486065104388231

Graph of the ZZ-function along the critical line