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Properties

Label 4-1323e2-1.1-c3e2-0-5
Degree 44
Conductor 17503291750329
Sign 11
Analytic cond. 6093.286093.28
Root an. cond. 8.835138.83513
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  − 2·2-s − 5·4-s − 22·5-s + 12·8-s + 44·10-s + 62·11-s + 64·13-s − 11·16-s + 32·17-s − 162·19-s + 110·20-s − 124·22-s + 170·23-s + 115·25-s − 128·26-s − 128·29-s − 178·31-s + 122·32-s − 64·34-s − 350·37-s + 324·38-s − 264·40-s − 58·41-s + 756·43-s − 310·44-s − 340·46-s + 180·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 5/8·4-s − 1.96·5-s + 0.530·8-s + 1.39·10-s + 1.69·11-s + 1.36·13-s − 0.171·16-s + 0.456·17-s − 1.95·19-s + 1.22·20-s − 1.20·22-s + 1.54·23-s + 0.919·25-s − 0.965·26-s − 0.819·29-s − 1.03·31-s + 0.673·32-s − 0.322·34-s − 1.55·37-s + 1.38·38-s − 1.04·40-s − 0.220·41-s + 2.68·43-s − 1.06·44-s − 1.08·46-s + 0.558·47-s + ⋯

Functional equation

Λ(s)=(1750329s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1750329s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 17503291750329    =    36⋅743^{6} \cdot 7^{4}
Sign: 11
Analytic conductor: 6093.286093.28
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1750329, ( :3/2,3/2), 1)(4,\ 1750329,\ (\ :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)
bad3 1 1
7 1 1
good2D4D_{4} 1+pT+9T2+p4T3+p6T4 1 + p T + 9 T^{2} + p^{4} T^{3} + p^{6} T^{4}
5D4D_{4} 1+22T+369T2+22p3T3+p6T4 1 + 22 T + 369 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−62T+3551T2−62p3T3+p6T4 1 - 62 T + 3551 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−64T+4840T2−64p3T3+p6T4 1 - 64 T + 4840 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−32T+9434T2−32p3T3+p6T4 1 - 32 T + 9434 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+162T+19311T2+162p3T3+p6T4 1 + 162 T + 19311 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−170T+24117T2−170p3T3+p6T4 1 - 170 T + 24117 T^{2} - 170 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+128T+52152T2+128p3T3+p6T4 1 + 128 T + 52152 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+178T+39181T2+178p3T3+p6T4 1 + 178 T + 39181 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+350T+129753T2+350p3T3+p6T4 1 + 350 T + 129753 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+58T+128315T2+58p3T3+p6T4 1 + 58 T + 128315 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−756T+288450T2−756p3T3+p6T4 1 - 756 T + 288450 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−180T+104354T2−180p3T3+p6T4 1 - 180 T + 104354 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−1088T+592538T2−1088p3T3+p6T4 1 - 1088 T + 592538 T^{2} - 1088 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+508T+174186T2+508p3T3+p6T4 1 + 508 T + 174186 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−152T+459666T2−152p3T3+p6T4 1 - 152 T + 459666 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−468T+88104T2−468p3T3+p6T4 1 - 468 T + 88104 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−14T+193629T2−14p3T3+p6T4 1 - 14 T + 193629 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+788T+930382T2+788p3T3+p6T4 1 + 788 T + 930382 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+476T+940570T2+476p3T3+p6T4 1 + 476 T + 940570 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+1492T+1690008T2+1492p3T3+p6T4 1 + 1492 T + 1690008 T^{2} + 1492 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−230T−8269T2−230p3T3+p6T4 1 - 230 T - 8269 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−640T+1770946T2−640p3T3+p6T4 1 - 640 T + 1770946 T^{2} - 640 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

−8.825598914480012675049488847398, −8.808765367792287276066199686761, −8.225301418840038481317918372500, −8.204868214447504008470033654239, −7.38208223884772472850539389718, −7.11735733582857931315557282028, −6.86058063298896491914217281851, −6.25286433625253564777466875682, −5.67431819133354653054082283609, −5.39215020820646574515783636867, −4.34574248222921015659893481514, −4.25101284896703264319687919152, −3.84166157028382341677910983629, −3.75639303810444641708480717361, −2.96470147417517952847425134780, −2.15682772641896408220025382330, −1.24380007062860037366932254682, −1.00912637869304958416781753381, 0, 0, 1.00912637869304958416781753381, 1.24380007062860037366932254682, 2.15682772641896408220025382330, 2.96470147417517952847425134780, 3.75639303810444641708480717361, 3.84166157028382341677910983629, 4.25101284896703264319687919152, 4.34574248222921015659893481514, 5.39215020820646574515783636867, 5.67431819133354653054082283609, 6.25286433625253564777466875682, 6.86058063298896491914217281851, 7.11735733582857931315557282028, 7.38208223884772472850539389718, 8.204868214447504008470033654239, 8.225301418840038481317918372500, 8.808765367792287276066199686761, 8.825598914480012675049488847398

Graph of the ZZ-function along the critical line