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Properties

Label 4-350e2-1.1-c3e2-0-0
Degree 44
Conductor 122500122500
Sign 11
Analytic cond. 426.450426.450
Root an. cond. 4.544304.54430
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  − 2·2-s + 10·3-s − 20·6-s − 28·7-s + 8·8-s + 27·9-s − 53·11-s − 50·13-s + 56·14-s − 16·16-s + 14·17-s − 54·18-s + 95·19-s − 280·21-s + 106·22-s + 23-s + 80·24-s + 100·26-s − 190·27-s − 412·29-s − 108·31-s − 530·33-s − 28·34-s − 57·37-s − 190·38-s − 500·39-s + 486·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.92·3-s − 1.36·6-s − 1.51·7-s + 0.353·8-s + 9-s − 1.45·11-s − 1.06·13-s + 1.06·14-s − 1/4·16-s + 0.199·17-s − 0.707·18-s + 1.14·19-s − 2.90·21-s + 1.02·22-s + 0.00906·23-s + 0.680·24-s + 0.754·26-s − 1.35·27-s − 2.63·29-s − 0.625·31-s − 2.79·33-s − 0.141·34-s − 0.253·37-s − 0.811·38-s − 2.05·39-s + 1.85·41-s + ⋯

Functional equation

Λ(s)=(122500s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(122500s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 122500122500    =    22⋅54⋅722^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 426.450426.450
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 122500, ( :3/2,3/2), 1)(4,\ 122500,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.010429481990.01042948199
L(12)L(\frac12) ≈\approx 0.010429481990.01042948199
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)
bad2C2C_2 1+pT+p2T2 1 + p T + p^{2} T^{2}
5 1 1
7C2C_2 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
good3C22C_2^2 1−10T+73T2−10p3T3+p6T4 1 - 10 T + 73 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1+53T+1478T2+53p3T3+p6T4 1 + 53 T + 1478 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4}
13C2C_2 (1+25T+p3T2)2 ( 1 + 25 T + p^{3} T^{2} )^{2}
17C22C_2^2 1−14T−4717T2−14p3T3+p6T4 1 - 14 T - 4717 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1−5pT+6p2T2−5p4T3+p6T4 1 - 5 p T + 6 p^{2} T^{2} - 5 p^{4} T^{3} + p^{6} T^{4}
23C22C_2^2 1−T−12166T2−p3T3+p6T4 1 - T - 12166 T^{2} - p^{3} T^{3} + p^{6} T^{4}
29C2C_2 (1+206T+p3T2)2 ( 1 + 206 T + p^{3} T^{2} )^{2}
31C22C_2^2 1+108T−18127T2+108p3T3+p6T4 1 + 108 T - 18127 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4}
37C22C_2^2 1+57T−47404T2+57p3T3+p6T4 1 + 57 T - 47404 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4}
41C2C_2 (1−243T+p3T2)2 ( 1 - 243 T + p^{3} T^{2} )^{2}
43C2C_2 (1+434T+p3T2)2 ( 1 + 434 T + p^{3} T^{2} )^{2}
47C22C_2^2 1+231T−50462T2+231p3T3+p6T4 1 + 231 T - 50462 T^{2} + 231 p^{3} T^{3} + p^{6} T^{4}
53C22C_2^2 1−263T−79708T2−263p3T3+p6T4 1 - 263 T - 79708 T^{2} - 263 p^{3} T^{3} + p^{6} T^{4}
59C22C_2^2 1+24T−204803T2+24p3T3+p6T4 1 + 24 T - 204803 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1+116T−213525T2+116p3T3+p6T4 1 + 116 T - 213525 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1+204T−259147T2+204p3T3+p6T4 1 + 204 T - 259147 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−484T+p3T2)2 ( 1 - 484 T + p^{3} T^{2} )^{2}
73C22C_2^2 1+692T+89847T2+692p3T3+p6T4 1 + 692 T + 89847 T^{2} + 692 p^{3} T^{3} + p^{6} T^{4}
79C22C_2^2 1+466T−275883T2+466p3T3+p6T4 1 + 466 T - 275883 T^{2} + 466 p^{3} T^{3} + p^{6} T^{4}
83C2C_2 (1+228T+p3T2)2 ( 1 + 228 T + p^{3} T^{2} )^{2}
89C22C_2^2 1−362T−573925T2−362p3T3+p6T4 1 - 362 T - 573925 T^{2} - 362 p^{3} T^{3} + p^{6} T^{4}
97C2C_2 (1+854T+p3T2)2 ( 1 + 854 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

−11.62653449852212042446102139838, −10.59944741008033365770995609372, −9.980884912876056519736665453697, −9.716601862681214091843336984543, −9.368383000616132148751032634179, −9.192871351758528600865688635038, −8.367380980222462789646690778089, −8.231485080810686994598404254350, −7.52470400170035615181889654244, −7.38677102647184090540279533039, −6.87663034429098675884645744904, −5.81600889575501099368160945527, −5.52145120538479514124072895137, −4.86069098557007738511345919304, −3.70778416748934452606666920858, −3.49916583501781047638451783395, −2.80864797526560914737190981383, −2.49470828326172294067253878203, −1.68917098489123767123829062158, −0.03314158824316176183718359528, 0.03314158824316176183718359528, 1.68917098489123767123829062158, 2.49470828326172294067253878203, 2.80864797526560914737190981383, 3.49916583501781047638451783395, 3.70778416748934452606666920858, 4.86069098557007738511345919304, 5.52145120538479514124072895137, 5.81600889575501099368160945527, 6.87663034429098675884645744904, 7.38677102647184090540279533039, 7.52470400170035615181889654244, 8.231485080810686994598404254350, 8.367380980222462789646690778089, 9.192871351758528600865688635038, 9.368383000616132148751032634179, 9.716601862681214091843336984543, 9.980884912876056519736665453697, 10.59944741008033365770995609372, 11.62653449852212042446102139838

Graph of the ZZ-function along the critical line