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Properties

Label 4-1575e2-1.1-c3e2-0-6
Degree 44
Conductor 24806252480625
Sign 11
Analytic cond. 8635.618635.61
Root an. cond. 9.639919.63991
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  + 2-s + 4-s + 14·7-s + 9·8-s + 22·11-s + 22·13-s + 14·14-s − 47·16-s + 116·17-s + 102·19-s + 22·22-s + 260·23-s + 22·26-s + 14·28-s + 196·29-s + 150·31-s − 103·32-s + 116·34-s + 96·37-s + 102·38-s + 176·41-s + 344·43-s + 22·44-s + 260·46-s + 560·47-s + 147·49-s + 22·52-s + ⋯
L(s)  = 1  + 0.353·2-s + 1/8·4-s + 0.755·7-s + 0.397·8-s + 0.603·11-s + 0.469·13-s + 0.267·14-s − 0.734·16-s + 1.65·17-s + 1.23·19-s + 0.213·22-s + 2.35·23-s + 0.165·26-s + 0.0944·28-s + 1.25·29-s + 0.869·31-s − 0.568·32-s + 0.585·34-s + 0.426·37-s + 0.435·38-s + 0.670·41-s + 1.21·43-s + 0.0753·44-s + 0.833·46-s + 1.73·47-s + 3/7·49-s + 0.0586·52-s + ⋯

Functional equation

Λ(s)=(2480625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2480625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 24806252480625    =    34⋅54⋅723^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 8635.618635.61
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2480625, ( :3/2,3/2), 1)(4,\ 2480625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 9.3558883829.355888382
L(12)L(\frac12) ≈\approx 9.3558883829.355888382
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)
bad3 1 1
5 1 1
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
good2D4D_{4} 1−T−p3T3+p6T4 1 - T - p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−2pT+2718T2−2p4T3+p6T4 1 - 2 p T + 2718 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4}
13D4D_{4} 1−22T+4450T2−22p3T3+p6T4 1 - 22 T + 4450 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−116T+9030T2−116p3T3+p6T4 1 - 116 T + 9030 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−102T+13134T2−102p3T3+p6T4 1 - 102 T + 13134 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−260T+34734T2−260p3T3+p6T4 1 - 260 T + 34734 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−196T+20942T2−196p3T3+p6T4 1 - 196 T + 20942 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−150T+36542T2−150p3T3+p6T4 1 - 150 T + 36542 T^{2} - 150 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−96T+82550T2−96p3T3+p6T4 1 - 96 T + 82550 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−176T−16914T2−176p3T3+p6T4 1 - 176 T - 16914 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−8pT+171958T2−8p4T3+p6T4 1 - 8 p T + 171958 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4}
47D4D_{4} 1−560T+248606T2−560p3T3+p6T4 1 - 560 T + 248606 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−326T+204138T2−326p3T3+p6T4 1 - 326 T + 204138 T^{2} - 326 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−844T+474182T2−844p3T3+p6T4 1 - 844 T + 474182 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+204T+455006T2+204p3T3+p6T4 1 + 204 T + 455006 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−104T+537670T2−104p3T3+p6T4 1 - 104 T + 537670 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+1670T+1384382T2+1670p3T3+p6T4 1 + 1670 T + 1384382 T^{2} + 1670 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−386T+152218T2−386p3T3+p6T4 1 - 386 T + 152218 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+888T+1007454T2+888p3T3+p6T4 1 + 888 T + 1007454 T^{2} + 888 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−928T+600710T2−928p3T3+p6T4 1 - 928 T + 600710 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+588T+1495334T2+588p3T3+p6T4 1 + 588 T + 1495334 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+522T+291282T2+522p3T3+p6T4 1 + 522 T + 291282 T^{2} + 522 p^{3} T^{3} + 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

−9.153353705586358740942453497971, −8.960656834628883649582115361595, −8.368641941146238382480236027923, −8.186452121397549362486943745535, −7.43914919027476010469579570411, −7.24818549978646837826075463789, −7.06808266693983678354571484620, −6.43959070042050899162803219312, −5.76931935054066266328814399972, −5.66835861805718313452857892909, −5.08914932039572729156213172901, −4.71682878057123809705274692271, −4.24060652882825234153686283219, −3.88725987228588153584368358120, −3.16535486675948784249367220519, −2.81147344313458567046704702605, −2.36092577727575558637389361475, −1.37364563684643671998184723910, −1.01387528047532492512745754127, −0.825756224556272554430617877710, 0.825756224556272554430617877710, 1.01387528047532492512745754127, 1.37364563684643671998184723910, 2.36092577727575558637389361475, 2.81147344313458567046704702605, 3.16535486675948784249367220519, 3.88725987228588153584368358120, 4.24060652882825234153686283219, 4.71682878057123809705274692271, 5.08914932039572729156213172901, 5.66835861805718313452857892909, 5.76931935054066266328814399972, 6.43959070042050899162803219312, 7.06808266693983678354571484620, 7.24818549978646837826075463789, 7.43914919027476010469579570411, 8.186452121397549362486943745535, 8.368641941146238382480236027923, 8.960656834628883649582115361595, 9.153353705586358740942453497971

Graph of the ZZ-function along the critical line