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Properties

Label 4-1232e2-1.1-c3e2-0-4
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  + 5·3-s − 17·5-s + 14·7-s − 21·9-s + 22·11-s − 86·13-s − 85·15-s − 66·17-s + 136·19-s + 70·21-s − 17·23-s + 95·25-s − 200·27-s + 124·29-s + 71·31-s + 110·33-s − 238·35-s − 135·37-s − 430·39-s − 618·41-s − 604·43-s + 357·45-s − 2·47-s + 147·49-s − 330·51-s + 152·53-s − 374·55-s + ⋯
L(s)  = 1  + 0.962·3-s − 1.52·5-s + 0.755·7-s − 7/9·9-s + 0.603·11-s − 1.83·13-s − 1.46·15-s − 0.941·17-s + 1.64·19-s + 0.727·21-s − 0.154·23-s + 0.759·25-s − 1.42·27-s + 0.794·29-s + 0.411·31-s + 0.580·33-s − 1.14·35-s − 0.599·37-s − 1.76·39-s − 2.35·41-s − 2.14·43-s + 1.18·45-s − 0.00620·47-s + 3/7·49-s − 0.906·51-s + 0.393·53-s − 0.916·55-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} 1−5T+46T2−5p3T3+p6T4 1 - 5 T + 46 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4}
5D4D_{4} 1+17T+194T2+17p3T3+p6T4 1 + 17 T + 194 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+86T+6186T2+86p3T3+p6T4 1 + 86 T + 6186 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+66T+1282T2+66p3T3+p6T4 1 + 66 T + 1282 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−136T+18114T2−136p3T3+p6T4 1 - 136 T + 18114 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+17T+18122T2+17p3T3+p6T4 1 + 17 T + 18122 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−124T+19790T2−124p3T3+p6T4 1 - 124 T + 19790 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−71T+59688T2−71p3T3+p6T4 1 - 71 T + 59688 T^{2} - 71 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+135T+101744T2+135p3T3+p6T4 1 + 135 T + 101744 T^{2} + 135 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+618T+203170T2+618p3T3+p6T4 1 + 618 T + 203170 T^{2} + 618 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+604T+211686T2+604p3T3+p6T4 1 + 604 T + 211686 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+2T−48226T2+2p3T3+p6T4 1 + 2 T - 48226 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1−152T−118042T2−152p3T3+p6T4 1 - 152 T - 118042 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−343T+251714T2−343p3T3+p6T4 1 - 343 T + 251714 T^{2} - 343 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−380T+7614T2−380p3T3+p6T4 1 - 380 T + 7614 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+1027T+851514T2+1027p3T3+p6T4 1 + 1027 T + 851514 T^{2} + 1027 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−1169T+1028606T2−1169p3T3+p6T4 1 - 1169 T + 1028606 T^{2} - 1169 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+94T+755106T2+94p3T3+p6T4 1 + 94 T + 755106 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+842T+1127694T2+842p3T3+p6T4 1 + 842 T + 1127694 T^{2} + 842 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−1486T+1670486T2−1486p3T3+p6T4 1 - 1486 T + 1670486 T^{2} - 1486 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+361T+1240724T2+361p3T3+p6T4 1 + 361 T + 1240724 T^{2} + 361 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−225T+1794896T2−225p3T3+p6T4 1 - 225 T + 1794896 T^{2} - 225 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

−8.965789640618317442242401539433, −8.596240282729508514910398704147, −8.194614939414958622820808412601, −8.183394936218209261580222839283, −7.47090262732493261806893103095, −7.36086694228590903178907474779, −6.64834927222021733116547255860, −6.61466513724391250280673592379, −5.47569522684061874972217817267, −5.26577134860883502587812435887, −4.74939898267623058648779762062, −4.46256433401156783300579823659, −3.57989679136885331387883811667, −3.54453773665103391522966030240, −2.91553962195133858767669820248, −2.40806162750964078785910804729, −1.86128630814384674719530421905, −1.06027715211277279575475315198, 0, 0, 1.06027715211277279575475315198, 1.86128630814384674719530421905, 2.40806162750964078785910804729, 2.91553962195133858767669820248, 3.54453773665103391522966030240, 3.57989679136885331387883811667, 4.46256433401156783300579823659, 4.74939898267623058648779762062, 5.26577134860883502587812435887, 5.47569522684061874972217817267, 6.61466513724391250280673592379, 6.64834927222021733116547255860, 7.36086694228590903178907474779, 7.47090262732493261806893103095, 8.183394936218209261580222839283, 8.194614939414958622820808412601, 8.596240282729508514910398704147, 8.965789640618317442242401539433

Graph of the ZZ-function along the critical line