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Properties

Label 4-1232e2-1.1-c3e2-0-3
Degree 44
Conductor 15178241517824
Sign 11
Analytic cond. 5283.885283.88
Root an. cond. 8.525868.52586
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  + 2·3-s − 8·5-s + 14·7-s − 6·9-s − 22·11-s − 18·13-s − 16·15-s − 74·17-s − 160·19-s + 28·21-s + 20·23-s − 182·25-s + 22·27-s + 72·29-s − 238·31-s − 44·33-s − 112·35-s − 56·37-s − 36·39-s + 262·41-s + 376·43-s + 48·45-s + 78·47-s + 147·49-s − 148·51-s + 496·53-s + 176·55-s + ⋯
L(s)  = 1  + 0.384·3-s − 0.715·5-s + 0.755·7-s − 2/9·9-s − 0.603·11-s − 0.384·13-s − 0.275·15-s − 1.05·17-s − 1.93·19-s + 0.290·21-s + 0.181·23-s − 1.45·25-s + 0.156·27-s + 0.461·29-s − 1.37·31-s − 0.232·33-s − 0.540·35-s − 0.248·37-s − 0.147·39-s + 0.997·41-s + 1.33·43-s + 0.159·45-s + 0.242·47-s + 3/7·49-s − 0.406·51-s + 1.28·53-s + 0.431·55-s + ⋯

Functional equation

Λ(s)=(1517824s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1517824s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 15178241517824    =    28⋅72⋅1122^{8} \cdot 7^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 5283.885283.88
Root analytic conductor: 8.525868.52586
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1517824, ( :3/2,3/2), 1)(4,\ 1517824,\ (\ :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
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
11C1C_1 (1+pT)2 ( 1 + p T )^{2}
good3D4D_{4} 1−2T+10T2−2p3T3+p6T4 1 - 2 T + 10 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1+8T+246T2+8p3T3+p6T4 1 + 8 T + 246 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+18T+3030T2+18p3T3+p6T4 1 + 18 T + 3030 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+74T+8070T2+74p3T3+p6T4 1 + 74 T + 8070 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+160T+20038T2+160p3T3+p6T4 1 + 160 T + 20038 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−20T+19934T2−20p3T3+p6T4 1 - 20 T + 19934 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−72T+30854T2−72p3T3+p6T4 1 - 72 T + 30854 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+238T+73338T2+238p3T3+p6T4 1 + 238 T + 73338 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+56T−42410T2+56p3T3+p6T4 1 + 56 T - 42410 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−262T+128358T2−262p3T3+p6T4 1 - 262 T + 128358 T^{2} - 262 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−376T+152038T2−376p3T3+p6T4 1 - 376 T + 152038 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−78T+171322T2−78p3T3+p6T4 1 - 78 T + 171322 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−496T+358758T2−496p3T3+p6T4 1 - 496 T + 358758 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+998T+620154T2+998p3T3+p6T4 1 + 998 T + 620154 T^{2} + 998 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−1562T+1063518T2−1562p3T3+p6T4 1 - 1562 T + 1063518 T^{2} - 1562 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−500T+658246T2−500p3T3+p6T4 1 - 500 T + 658246 T^{2} - 500 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+2100T+1817342T2+2100p3T3+p6T4 1 + 2100 T + 1817342 T^{2} + 2100 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−586T+43758T2−586p3T3+p6T4 1 - 586 T + 43758 T^{2} - 586 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−16T+985422T2−16p3T3+p6T4 1 - 16 T + 985422 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−1648T+1419270T2−1648p3T3+p6T4 1 - 1648 T + 1419270 T^{2} - 1648 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+940T+1238838T2+940p3T3+p6T4 1 + 940 T + 1238838 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1152T+2072622T2−1152p3T3+p6T4 1 - 1152 T + 2072622 T^{2} - 1152 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

−8.959972410253618877614663164912, −8.766682748563349426969670543923, −8.197669088511216850279697691680, −8.097049928413454484847070384127, −7.46976553004778758148066027127, −7.28692272837988702616436077594, −6.73993681424293674805772528041, −6.22311741924035161602309228244, −5.68962961358465638430633337136, −5.40345100893646698679185188488, −4.61738634343120990046799781716, −4.41699164947124921496560406802, −3.92947039733666461026610878244, −3.57671291501064408563487291310, −2.56342755547118177495019595074, −2.39328452904396865660747681450, −1.93618153059689036238298678767, −1.04459560422884113672584049805, 0, 0, 1.04459560422884113672584049805, 1.93618153059689036238298678767, 2.39328452904396865660747681450, 2.56342755547118177495019595074, 3.57671291501064408563487291310, 3.92947039733666461026610878244, 4.41699164947124921496560406802, 4.61738634343120990046799781716, 5.40345100893646698679185188488, 5.68962961358465638430633337136, 6.22311741924035161602309228244, 6.73993681424293674805772528041, 7.28692272837988702616436077594, 7.46976553004778758148066027127, 8.097049928413454484847070384127, 8.197669088511216850279697691680, 8.766682748563349426969670543923, 8.959972410253618877614663164912

Graph of the ZZ-function along the critical line