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Properties

Label 4-2028e2-1.1-c3e2-0-2
Degree 44
Conductor 41127844112784
Sign 11
Analytic cond. 14317.514317.5
Root an. cond. 10.938710.9387
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 27·9-s + 28·17-s − 144·23-s + 246·25-s + 108·27-s + 204·29-s + 280·43-s − 338·49-s + 168·51-s + 1.05e3·53-s − 820·61-s − 864·69-s + 1.47e3·75-s − 1.28e3·79-s + 405·81-s + 1.22e3·87-s + 1.22e3·101-s + 2.70e3·103-s − 3.03e3·113-s − 1.96e3·121-s + 127-s + 1.68e3·129-s + 131-s + 137-s + 139-s − 2.02e3·147-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 0.399·17-s − 1.30·23-s + 1.96·25-s + 0.769·27-s + 1.30·29-s + 0.993·43-s − 0.985·49-s + 0.461·51-s + 2.72·53-s − 1.72·61-s − 1.50·69-s + 2.27·75-s − 1.82·79-s + 5/9·81-s + 1.50·87-s + 1.20·101-s + 2.58·103-s − 2.52·113-s − 1.47·121-s + 0.000698·127-s + 1.14·129-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 1.13·147-s + ⋯

Functional equation

Λ(s)=(4112784s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(4112784s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4112784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 41127844112784    =    24⋅32⋅1342^{4} \cdot 3^{2} \cdot 13^{4}
Sign: 11
Analytic conductor: 14317.514317.5
Root analytic conductor: 10.938710.9387
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4112784, ( :3/2,3/2), 1)(4,\ 4112784,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.3249115515.324911551
L(12)L(\frac12) ≈\approx 5.3249115515.324911551
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
3C1C_1 (1−pT)2 ( 1 - p T )^{2}
13 1 1
good5C22C_2^2 1−246T2+p6T4 1 - 246 T^{2} + p^{6} T^{4}
7C22C_2^2 1+338T2+p6T4 1 + 338 T^{2} + p^{6} T^{4}
11C22C_2^2 1+1962T2+p6T4 1 + 1962 T^{2} + p^{6} T^{4}
17C2C_2 (1−14T+p3T2)2 ( 1 - 14 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−13702T2+p6T4 1 - 13702 T^{2} + p^{6} T^{4}
23C2C_2 (1+72T+p3T2)2 ( 1 + 72 T + p^{3} T^{2} )^{2}
29C2C_2 (1−102T+p3T2)2 ( 1 - 102 T + p^{3} T^{2} )^{2}
31C22C_2^2 1−41086T2+p6T4 1 - 41086 T^{2} + p^{6} T^{4}
37C22C_2^2 1+47690T2+p6T4 1 + 47690 T^{2} + p^{6} T^{4}
41C22C_2^2 1−75342T2+p6T4 1 - 75342 T^{2} + p^{6} T^{4}
43C2C_2 (1−140T+p3T2)2 ( 1 - 140 T + p^{3} T^{2} )^{2}
47C22C_2^2 1−120030T2+p6T4 1 - 120030 T^{2} + p^{6} T^{4}
53C2C_2 (1−526T+p3T2)2 ( 1 - 526 T + p^{3} T^{2} )^{2}
59C22C_2^2 1−300534T2+p6T4 1 - 300534 T^{2} + p^{6} T^{4}
61C2C_2 (1+410T+p3T2)2 ( 1 + 410 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−246310T2+p6T4 1 - 246310 T^{2} + p^{6} T^{4}
71C22C_2^2 1+58578T2+p6T4 1 + 58578 T^{2} + p^{6} T^{4}
73C22C_2^2 1−521998T2+p6T4 1 - 521998 T^{2} + p^{6} T^{4}
79C2C_2 (1+640T+p3T2)2 ( 1 + 640 T + p^{3} T^{2} )^{2}
83C22C_2^2 1+760826T2+p6T4 1 + 760826 T^{2} + p^{6} T^{4}
89C22C_2^2 1+692562T2+p6T4 1 + 692562 T^{2} + p^{6} T^{4}
97C22C_2^2 1−1626430T2+p6T4 1 - 1626430 T^{2} + p^{6} 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

−9.124274325822656895203433760919, −8.519926980992739635935754671017, −8.374353947812951283536540023160, −7.71823997685475186750354013703, −7.64628119322417793580509004451, −7.06620956274989615360234939962, −6.75685636226184684741923616383, −6.22415003086038462786123877335, −5.96066261953728101922559712026, −5.23672851518401069917398648869, −4.94688989206704124506213762350, −4.39768572170333406033630841599, −4.02445682722962573778310429542, −3.60408325007648737536345349268, −2.99794688425899016482321795487, −2.64924450538313261881176546796, −2.33425926609340014814110981864, −1.48225111171507079653476834222, −1.16898147348237885581706773418, −0.44891388009551541113233175732, 0.44891388009551541113233175732, 1.16898147348237885581706773418, 1.48225111171507079653476834222, 2.33425926609340014814110981864, 2.64924450538313261881176546796, 2.99794688425899016482321795487, 3.60408325007648737536345349268, 4.02445682722962573778310429542, 4.39768572170333406033630841599, 4.94688989206704124506213762350, 5.23672851518401069917398648869, 5.96066261953728101922559712026, 6.22415003086038462786123877335, 6.75685636226184684741923616383, 7.06620956274989615360234939962, 7.64628119322417793580509004451, 7.71823997685475186750354013703, 8.374353947812951283536540023160, 8.519926980992739635935754671017, 9.124274325822656895203433760919

Graph of the ZZ-function along the critical line