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Properties

Label 4-8112e2-1.1-c1e2-0-5
Degree 44
Conductor 6580454465804544
Sign 11
Analytic cond. 4195.754195.75
Root an. cond. 8.048268.04826
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  + 2·3-s + 3·9-s − 12·23-s + 2·25-s + 4·27-s + 12·29-s − 2·43-s − 11·49-s + 24·53-s + 2·61-s − 24·69-s + 4·75-s + 22·79-s + 5·81-s + 24·87-s + 36·101-s − 2·103-s + 12·107-s + 12·113-s − 10·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 22·147-s + 149-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 2.50·23-s + 2/5·25-s + 0.769·27-s + 2.22·29-s − 0.304·43-s − 1.57·49-s + 3.29·53-s + 0.256·61-s − 2.88·69-s + 0.461·75-s + 2.47·79-s + 5/9·81-s + 2.57·87-s + 3.58·101-s − 0.197·103-s + 1.16·107-s + 1.12·113-s − 0.909·121-s + 0.0887·127-s − 0.352·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.81·147-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=(65804544s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(65804544s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6580454465804544    =    28⋅32⋅1342^{8} \cdot 3^{2} \cdot 13^{4}
Sign: 11
Analytic conductor: 4195.754195.75
Root analytic conductor: 8.048268.04826
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 65804544, ( :1/2,1/2), 1)(4,\ 65804544,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.5244962065.524496206
L(12)L(\frac12) ≈\approx 5.5244962065.524496206
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
3C1C_1 (1−T)2 ( 1 - T )^{2}
13 1 1
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C22C_2^2 1+59T2+p2T4 1 + 59 T^{2} + p^{2} T^{4}
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
53C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
59C22C_2^2 1+106T2+p2T4 1 + 106 T^{2} + p^{2} T^{4}
61C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
67C22C_2^2 1+59T2+p2T4 1 + 59 T^{2} + p^{2} T^{4}
71C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
73C22C_2^2 1+143T2+p2T4 1 + 143 T^{2} + p^{2} T^{4}
79C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
83C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
89C22C_2^2 1+130T2+p2T4 1 + 130 T^{2} + p^{2} T^{4}
97C22C_2^2 1+167T2+p2T4 1 + 167 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

−7.914742258493993233136385732467, −7.78573258930618977651538713247, −7.48636729143301806935570681592, −6.91902854097483798828469619614, −6.67378145344098308048578793339, −6.35644948737587482518629829947, −6.01124298476723272419499876157, −5.61078665261345951603692006958, −5.08143603512056640526757292127, −4.79907325999365744746282630217, −4.29965369160807394585883922723, −4.15786349006164698609965746421, −3.48934529834465575486508023284, −3.45678724041477860448606587734, −2.85567185138923397641951261937, −2.43042029089779197788124239944, −2.00579167201248311567823960584, −1.82581884991847559444437499937, −0.931919197640142301796455455128, −0.58609583419050332989630214613, 0.58609583419050332989630214613, 0.931919197640142301796455455128, 1.82581884991847559444437499937, 2.00579167201248311567823960584, 2.43042029089779197788124239944, 2.85567185138923397641951261937, 3.45678724041477860448606587734, 3.48934529834465575486508023284, 4.15786349006164698609965746421, 4.29965369160807394585883922723, 4.79907325999365744746282630217, 5.08143603512056640526757292127, 5.61078665261345951603692006958, 6.01124298476723272419499876157, 6.35644948737587482518629829947, 6.67378145344098308048578793339, 6.91902854097483798828469619614, 7.48636729143301806935570681592, 7.78573258930618977651538713247, 7.914742258493993233136385732467

Graph of the ZZ-function along the critical line