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Properties

Label 4-18e4-1.1-c3e2-0-5
Degree 44
Conductor 104976104976
Sign 11
Analytic cond. 365.445365.445
Root an. cond. 4.372254.37225
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  − 9·5-s + 7-s − 63·11-s + 28·13-s + 144·17-s + 196·19-s − 126·23-s + 125·25-s + 126·29-s + 259·31-s − 9·35-s + 772·37-s + 450·41-s + 34·43-s + 54·47-s + 343·49-s − 1.38e3·53-s + 567·55-s − 180·59-s + 280·61-s − 252·65-s + 586·67-s + 1.00e3·71-s + 322·73-s − 63·77-s − 440·79-s − 999·83-s + ⋯
L(s)  = 1  − 0.804·5-s + 0.0539·7-s − 1.72·11-s + 0.597·13-s + 2.05·17-s + 2.36·19-s − 1.14·23-s + 25-s + 0.806·29-s + 1.50·31-s − 0.0434·35-s + 3.43·37-s + 1.71·41-s + 0.120·43-s + 0.167·47-s + 49-s − 3.59·53-s + 1.39·55-s − 0.397·59-s + 0.587·61-s − 0.480·65-s + 1.06·67-s + 1.68·71-s + 0.516·73-s − 0.0932·77-s − 0.626·79-s − 1.32·83-s + ⋯

Functional equation

Λ(s)=(104976s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(104976s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 104976104976    =    24⋅382^{4} \cdot 3^{8}
Sign: 11
Analytic conductor: 365.445365.445
Root analytic conductor: 4.372254.37225
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 104976, ( :3/2,3/2), 1)(4,\ 104976,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7087340882.708734088
L(12)L(\frac12) ≈\approx 2.7087340882.708734088
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
good5C22C_2^2 1+9T−44T2+9p3T3+p6T4 1 + 9 T - 44 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4}
7C22C_2^2 1−T−342T2−p3T3+p6T4 1 - T - 342 T^{2} - p^{3} T^{3} + p^{6} T^{4}
11C22C_2^2 1+63T+2638T2+63p3T3+p6T4 1 + 63 T + 2638 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4}
13C22C_2^2 1−28T−1413T2−28p3T3+p6T4 1 - 28 T - 1413 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4}
17C2C_2 (1−72T+p3T2)2 ( 1 - 72 T + p^{3} T^{2} )^{2}
19C2C_2 (1−98T+p3T2)2 ( 1 - 98 T + p^{3} T^{2} )^{2}
23C22C_2^2 1+126T+3709T2+126p3T3+p6T4 1 + 126 T + 3709 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4}
29C22C_2^2 1−126T−8513T2−126p3T3+p6T4 1 - 126 T - 8513 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4}
31C22C_2^2 1−259T+37290T2−259p3T3+p6T4 1 - 259 T + 37290 T^{2} - 259 p^{3} T^{3} + p^{6} T^{4}
37C2C_2 (1−386T+p3T2)2 ( 1 - 386 T + p^{3} T^{2} )^{2}
41C22C_2^2 1−450T+133579T2−450p3T3+p6T4 1 - 450 T + 133579 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4}
43C22C_2^2 1−34T−78351T2−34p3T3+p6T4 1 - 34 T - 78351 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4}
47C22C_2^2 1−54T−100907T2−54p3T3+p6T4 1 - 54 T - 100907 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4}
53C2C_2 (1+693T+p3T2)2 ( 1 + 693 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+180T−172979T2+180p3T3+p6T4 1 + 180 T - 172979 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4}
61C22C_2^2 1−280T−148581T2−280p3T3+p6T4 1 - 280 T - 148581 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4}
67C22C_2^2 1−586T+42633T2−586p3T3+p6T4 1 - 586 T + 42633 T^{2} - 586 p^{3} T^{3} + p^{6} T^{4}
71C2C_2 (1−504T+p3T2)2 ( 1 - 504 T + p^{3} T^{2} )^{2}
73C2C_2 (1−161T+p3T2)2 ( 1 - 161 T + p^{3} T^{2} )^{2}
79C22C_2^2 1+440T−299439T2+440p3T3+p6T4 1 + 440 T - 299439 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4}
83C22C_2^2 1+999T+426214T2+999p3T3+p6T4 1 + 999 T + 426214 T^{2} + 999 p^{3} T^{3} + p^{6} T^{4}
89C2C_2 (1−882T+p3T2)2 ( 1 - 882 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−721T−392832T2−721p3T3+p6T4 1 - 721 T - 392832 T^{2} - 721 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

−11.35101465547225160168113612762, −11.14535436853415931281237418873, −10.35843789578670937672571032185, −10.10994286260179528909852532421, −9.536542371664218804918650587077, −9.317981175865497587191162646416, −8.200476841792196676564309207664, −7.982510118586333691003602260837, −7.63574343568576525627688197148, −7.59525428254638429934892376938, −6.37967877519881137023231582662, −6.07905970095647360429010729322, −5.35994749583048097088568535573, −5.00833592551767471313979028454, −4.31940513893412940706157992357, −3.60537207784656534884930927804, −2.87388971897964494457200464816, −2.68655719901948491506880040306, −1.06383263312562327872496585481, −0.77402054066306260397507537847, 0.77402054066306260397507537847, 1.06383263312562327872496585481, 2.68655719901948491506880040306, 2.87388971897964494457200464816, 3.60537207784656534884930927804, 4.31940513893412940706157992357, 5.00833592551767471313979028454, 5.35994749583048097088568535573, 6.07905970095647360429010729322, 6.37967877519881137023231582662, 7.59525428254638429934892376938, 7.63574343568576525627688197148, 7.982510118586333691003602260837, 8.200476841792196676564309207664, 9.317981175865497587191162646416, 9.536542371664218804918650587077, 10.10994286260179528909852532421, 10.35843789578670937672571032185, 11.14535436853415931281237418873, 11.35101465547225160168113612762

Graph of the ZZ-function along the critical line