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Properties

Label 4-1872e2-1.1-c3e2-0-9
Degree 44
Conductor 35043843504384
Sign 11
Analytic cond. 12199.512199.5
Root an. cond. 10.509510.5095
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  − 12·5-s − 44·7-s + 52·11-s − 26·13-s + 20·17-s + 60·19-s − 8·23-s + 30·25-s − 132·29-s + 140·31-s + 528·35-s + 68·37-s − 28·41-s + 36·47-s + 938·49-s − 668·53-s − 624·55-s − 508·59-s + 340·61-s + 312·65-s + 940·67-s + 300·71-s + 1.12e3·73-s − 2.28e3·77-s + 1.52e3·79-s + 524·83-s − 240·85-s + ⋯
L(s)  = 1  − 1.07·5-s − 2.37·7-s + 1.42·11-s − 0.554·13-s + 0.285·17-s + 0.724·19-s − 0.0725·23-s + 6/25·25-s − 0.845·29-s + 0.811·31-s + 2.54·35-s + 0.302·37-s − 0.106·41-s + 0.111·47-s + 2.73·49-s − 1.73·53-s − 1.52·55-s − 1.12·59-s + 0.713·61-s + 0.595·65-s + 1.71·67-s + 0.501·71-s + 1.80·73-s − 3.38·77-s + 2.16·79-s + 0.692·83-s − 0.306·85-s + ⋯

Functional equation

Λ(s)=(3504384s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(3504384s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 35043843504384    =    28⋅34⋅1322^{8} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 12199.512199.5
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 3504384, ( :3/2,3/2), 1)(4,\ 3504384,\ (\ :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)
bad2 1 1
3 1 1
13C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1+12T+114T2+12p3T3+p6T4 1 + 12 T + 114 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1+44T+998T2+44p3T3+p6T4 1 + 44 T + 998 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4}
11C2C_2 (1−26T+p3T2)2 ( 1 - 26 T + p^{3} T^{2} )^{2}
17D4D_{4} 1−20T+9238T2−20p3T3+p6T4 1 - 20 T + 9238 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−60T+10318T2−60p3T3+p6T4 1 - 60 T + 10318 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+8T−418T2+8p3T3+p6T4 1 + 8 T - 418 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+132T+28366T2+132p3T3+p6T4 1 + 132 T + 28366 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−140T+25782T2−140p3T3+p6T4 1 - 140 T + 25782 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−68T+68750T2−68p3T3+p6T4 1 - 68 T + 68750 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+28T+129610T2+28p3T3+p6T4 1 + 28 T + 129610 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1+2914pT2+p6T4 1 + 2914 p T^{2} + p^{6} T^{4}
47D4D_{4} 1−36T+201778T2−36p3T3+p6T4 1 - 36 T + 201778 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+668T+406558T2+668p3T3+p6T4 1 + 668 T + 406558 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+508T+392026T2+508p3T3+p6T4 1 + 508 T + 392026 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−340T+471854T2−340p3T3+p6T4 1 - 340 T + 471854 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−940T+504398T2−940p3T3+p6T4 1 - 940 T + 504398 T^{2} - 940 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−300T−57006T2−300p3T3+p6T4 1 - 300 T - 57006 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−1124T+1087686T2−1124p3T3+p6T4 1 - 1124 T + 1087686 T^{2} - 1124 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1520T+1230686T2−1520p3T3+p6T4 1 - 1520 T + 1230686 T^{2} - 1520 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−524T+908810T2−524p3T3+p6T4 1 - 524 T + 908810 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+1900T+2312266T2+1900p3T3+p6T4 1 + 1900 T + 2312266 T^{2} + 1900 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1436T+2285142T2+1436p3T3+p6T4 1 + 1436 T + 2285142 T^{2} + 1436 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

−8.659314656201227123907407454359, −8.281070193100072734471404005118, −7.78710342068889981386247026438, −7.58492111603631264122147637485, −6.84131486050085617790809602983, −6.77124169218390644567232598628, −6.42864839786140833994736941152, −6.09929419067110008098845112542, −5.35483764183193332685489887111, −5.16319157197850804973390641030, −4.31791768881848020165286053694, −3.99440462185048265865660132166, −3.55414415699425161581413133450, −3.40417562057635315385649364014, −2.78038519231245257871463135579, −2.35075911058161871325755520422, −1.36369030630018271198301901855, −0.884414296797356043782937990923, 0, 0, 0.884414296797356043782937990923, 1.36369030630018271198301901855, 2.35075911058161871325755520422, 2.78038519231245257871463135579, 3.40417562057635315385649364014, 3.55414415699425161581413133450, 3.99440462185048265865660132166, 4.31791768881848020165286053694, 5.16319157197850804973390641030, 5.35483764183193332685489887111, 6.09929419067110008098845112542, 6.42864839786140833994736941152, 6.77124169218390644567232598628, 6.84131486050085617790809602983, 7.58492111603631264122147637485, 7.78710342068889981386247026438, 8.281070193100072734471404005118, 8.659314656201227123907407454359

Graph of the ZZ-function along the critical line