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Properties

Label 4-48e4-1.1-c3e2-0-32
Degree 44
Conductor 53084165308416
Sign 11
Analytic cond. 18479.718479.7
Root an. cond. 11.659311.6593
Motivic weight 33
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  + 12·5-s + 40·13-s − 16·17-s − 142·25-s − 92·29-s − 328·37-s − 624·41-s − 238·49-s + 532·53-s − 264·61-s + 480·65-s + 492·73-s − 192·85-s − 2.78e3·89-s − 604·97-s + 2.93e3·101-s − 3.12e3·109-s + 3.10e3·113-s − 870·121-s − 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.07·5-s + 0.853·13-s − 0.228·17-s − 1.13·25-s − 0.589·29-s − 1.45·37-s − 2.37·41-s − 0.693·49-s + 1.37·53-s − 0.554·61-s + 0.915·65-s + 0.788·73-s − 0.245·85-s − 3.31·89-s − 0.632·97-s + 2.88·101-s − 2.74·109-s + 2.58·113-s − 0.653·121-s − 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

Λ(s)=(5308416s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(5308416s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 53084165308416    =    216⋅342^{16} \cdot 3^{4}
Sign: 11
Analytic conductor: 18479.718479.7
Root analytic conductor: 11.659311.6593
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 5308416, ( :3/2,3/2), 1)(4,\ 5308416,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
good5C2C_2 (1−6T+p3T2)2 ( 1 - 6 T + p^{3} T^{2} )^{2}
7C22C_2^2 1+34pT2+p6T4 1 + 34 p T^{2} + p^{6} T^{4}
11C22C_2^2 1+870T2+p6T4 1 + 870 T^{2} + p^{6} T^{4}
13C2C_2 (1−20T+p3T2)2 ( 1 - 20 T + p^{3} T^{2} )^{2}
17C2C_2 (1+8T+p3T2)2 ( 1 + 8 T + p^{3} T^{2} )^{2}
19C22C_2^2 1+6550T2+p6T4 1 + 6550 T^{2} + p^{6} T^{4}
23C22C_2^2 1−4338T2+p6T4 1 - 4338 T^{2} + p^{6} T^{4}
29C2C_2 (1+46T+p3T2)2 ( 1 + 46 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+59134T2+p6T4 1 + 59134 T^{2} + p^{6} T^{4}
37C2C_2 (1+164T+p3T2)2 ( 1 + 164 T + p^{3} T^{2} )^{2}
41C2C_2 (1+312T+p3T2)2 ( 1 + 312 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−20186T2+p6T4 1 - 20186 T^{2} + p^{6} T^{4}
47C22C_2^2 1+178974T2+p6T4 1 + 178974 T^{2} + p^{6} T^{4}
53C2C_2 (1−266T+p3T2)2 ( 1 - 266 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+346246T2+p6T4 1 + 346246 T^{2} + p^{6} T^{4}
61C2C_2 (1+132T+p3T2)2 ( 1 + 132 T + p^{3} T^{2} )^{2}
67C22C_2^2 1+343478T2+p6T4 1 + 343478 T^{2} + p^{6} T^{4}
71C22C_2^2 1+257070T2+p6T4 1 + 257070 T^{2} + p^{6} T^{4}
73C2C_2 (1−246T+p3T2)2 ( 1 - 246 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+931870T2+p6T4 1 + 931870 T^{2} + p^{6} T^{4}
83C22C_2^2 1+195606T2+p6T4 1 + 195606 T^{2} + p^{6} T^{4}
89C2C_2 (1+1392T+p3T2)2 ( 1 + 1392 T + p^{3} T^{2} )^{2}
97C2C_2 (1+302T+p3T2)2 ( 1 + 302 T + p^{3} 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

−8.454335724385641876241301406715, −8.352853269697175036416224998437, −7.53322603716415626658945077888, −7.40548272531159595479192813363, −6.77919196205150001304123947919, −6.52183601045389796271688640757, −6.09546471330166877581861176464, −5.74306013495796717667003802687, −5.26164519607894616552902495729, −5.13467139041374179437178461999, −4.41536196509713758215398726909, −3.93630733941639700700117854415, −3.49327041316085531818569445788, −3.20871725173744602139833458062, −2.27203965202186773816555075488, −2.17941267894278403263094625231, −1.42295836020293354402691344283, −1.27183521030992646316492767112, 0, 0, 1.27183521030992646316492767112, 1.42295836020293354402691344283, 2.17941267894278403263094625231, 2.27203965202186773816555075488, 3.20871725173744602139833458062, 3.49327041316085531818569445788, 3.93630733941639700700117854415, 4.41536196509713758215398726909, 5.13467139041374179437178461999, 5.26164519607894616552902495729, 5.74306013495796717667003802687, 6.09546471330166877581861176464, 6.52183601045389796271688640757, 6.77919196205150001304123947919, 7.40548272531159595479192813363, 7.53322603716415626658945077888, 8.352853269697175036416224998437, 8.454335724385641876241301406715

Graph of the ZZ-function along the critical line