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Properties

Label 4-280e2-1.1-c3e2-0-0
Degree 44
Conductor 7840078400
Sign 11
Analytic cond. 272.928272.928
Root an. cond. 4.064544.06454
Motivic weight 33
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  − 7·3-s + 5·5-s + 7·7-s + 27·9-s − 58·11-s + 164·13-s − 35·15-s − 50·17-s − 64·19-s − 49·21-s + 111·23-s − 224·27-s + 206·29-s + 130·31-s + 406·33-s + 35·35-s − 376·37-s − 1.14e3·39-s − 614·41-s − 394·43-s + 135·45-s − 120·47-s − 294·49-s + 350·51-s + 508·53-s − 290·55-s + 448·57-s + ⋯
L(s)  = 1  − 1.34·3-s + 0.447·5-s + 0.377·7-s + 9-s − 1.58·11-s + 3.49·13-s − 0.602·15-s − 0.713·17-s − 0.772·19-s − 0.509·21-s + 1.00·23-s − 1.59·27-s + 1.31·29-s + 0.753·31-s + 2.14·33-s + 0.169·35-s − 1.67·37-s − 4.71·39-s − 2.33·41-s − 1.39·43-s + 0.447·45-s − 0.372·47-s − 6/7·49-s + 0.960·51-s + 1.31·53-s − 0.710·55-s + 1.04·57-s + ⋯

Functional equation

Λ(s)=(78400s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(78400s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 7840078400    =    26⋅52⋅722^{6} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 272.928272.928
Root analytic conductor: 4.064544.06454
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 78400, ( :3/2,3/2), 1)(4,\ 78400,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.3982153531.398215353
L(12)L(\frac12) ≈\approx 1.3982153531.398215353
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
5C2C_2 1−pT+p2T2 1 - p T + p^{2} T^{2}
7C2C_2 1−pT+p3T2 1 - p T + p^{3} T^{2}
good3C22C_2^2 1+7T+22T2+7p3T3+p6T4 1 + 7 T + 22 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1+58T+2033T2+58p3T3+p6T4 1 + 58 T + 2033 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4}
13C2C_2 (1−82T+p3T2)2 ( 1 - 82 T + p^{3} T^{2} )^{2}
17C22C_2^2 1+50T−2413T2+50p3T3+p6T4 1 + 50 T - 2413 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+64T−2763T2+64p3T3+p6T4 1 + 64 T - 2763 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4}
23C22C_2^2 1−111T+154T2−111p3T3+p6T4 1 - 111 T + 154 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4}
29C2C_2 (1−103T+p3T2)2 ( 1 - 103 T + p^{3} T^{2} )^{2}
31C22C_2^2 1−130T−12891T2−130p3T3+p6T4 1 - 130 T - 12891 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4}
37C22C_2^2 1+376T+90723T2+376p3T3+p6T4 1 + 376 T + 90723 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4}
41C2C_2 (1+307T+p3T2)2 ( 1 + 307 T + p^{3} T^{2} )^{2}
43C2C_2 (1+197T+p3T2)2 ( 1 + 197 T + p^{3} T^{2} )^{2}
47C22C_2^2 1+120T−89423T2+120p3T3+p6T4 1 + 120 T - 89423 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4}
53C22C_2^2 1−508T+109187T2−508p3T3+p6T4 1 - 508 T + 109187 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4}
59C22C_2^2 1+600T+154621T2+600p3T3+p6T4 1 + 600 T + 154621 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1−165T−199756T2−165p3T3+p6T4 1 - 165 T - 199756 T^{2} - 165 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1−633T+99926T2−633p3T3+p6T4 1 - 633 T + 99926 T^{2} - 633 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−840T+p3T2)2 ( 1 - 840 T + p^{3} T^{2} )^{2}
73C22C_2^2 1+606T−21781T2+606p3T3+p6T4 1 + 606 T - 21781 T^{2} + 606 p^{3} T^{3} + p^{6} T^{4}
79C22C_2^2 1−1316T+1238817T2−1316p3T3+p6T4 1 - 1316 T + 1238817 T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4}
83C2C_2 (1−61T+p3T2)2 ( 1 - 61 T + p^{3} T^{2} )^{2}
89C22C_2^2 1−187T−670000T2−187p3T3+p6T4 1 - 187 T - 670000 T^{2} - 187 p^{3} T^{3} + p^{6} T^{4}
97C2C_2 (1−406T+p3T2)2 ( 1 - 406 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

−11.59560752761809697768659222039, −10.99964615122909534875762087810, −10.89034930035001721152852817513, −10.36271321260575420694687390688, −10.18100299253983700917528019161, −9.248393436704561328086756911518, −8.612740752605213903175287420079, −8.284062543157981878091260833824, −8.080120498539775468298745083400, −6.76478147636842431553882527762, −6.73044293522881481754444403738, −6.15926143775471184016708559026, −5.67745049379818177983647645963, −4.95362632355586402340829423380, −4.89795716314073745381313288440, −3.69857540536067335720712030590, −3.36276871556921022259308770403, −2.08338995306755589223515431297, −1.41318428586734288907352820178, −0.52185977981391718091680818150, 0.52185977981391718091680818150, 1.41318428586734288907352820178, 2.08338995306755589223515431297, 3.36276871556921022259308770403, 3.69857540536067335720712030590, 4.89795716314073745381313288440, 4.95362632355586402340829423380, 5.67745049379818177983647645963, 6.15926143775471184016708559026, 6.73044293522881481754444403738, 6.76478147636842431553882527762, 8.080120498539775468298745083400, 8.284062543157981878091260833824, 8.612740752605213903175287420079, 9.248393436704561328086756911518, 10.18100299253983700917528019161, 10.36271321260575420694687390688, 10.89034930035001721152852817513, 10.99964615122909534875762087810, 11.59560752761809697768659222039

Graph of the ZZ-function along the critical line