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Properties

Label 4-1152e2-1.1-c3e2-0-3
Degree 44
Conductor 13271041327104
Sign 11
Analytic cond. 4619.944619.94
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s + 8·7-s − 92·11-s − 100·13-s − 92·17-s + 4·19-s + 8·23-s − 46·25-s + 84·29-s + 384·31-s + 32·35-s + 172·37-s + 300·41-s + 300·43-s − 16·47-s − 446·49-s − 12·53-s − 368·55-s + 644·59-s − 292·61-s − 400·65-s − 172·67-s + 408·71-s + 412·73-s − 736·77-s + 400·79-s − 948·83-s + ⋯
L(s)  = 1  + 0.357·5-s + 0.431·7-s − 2.52·11-s − 2.13·13-s − 1.31·17-s + 0.0482·19-s + 0.0725·23-s − 0.367·25-s + 0.537·29-s + 2.22·31-s + 0.154·35-s + 0.764·37-s + 1.14·41-s + 1.06·43-s − 0.0496·47-s − 1.30·49-s − 0.0311·53-s − 0.902·55-s + 1.42·59-s − 0.612·61-s − 0.763·65-s − 0.313·67-s + 0.681·71-s + 0.660·73-s − 1.08·77-s + 0.569·79-s − 1.25·83-s + ⋯

Functional equation

Λ(s)=(1327104s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1327104s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 13271041327104    =    214⋅342^{14} \cdot 3^{4}
Sign: 11
Analytic conductor: 4619.944619.94
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1327104, ( :3/2,3/2), 1)(4,\ 1327104,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.1622219601.162221960
L(12)L(\frac12) ≈\approx 1.1622219601.162221960
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
good5D4D_{4} 1−4T+62T2−4p3T3+p6T4 1 - 4 T + 62 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−8T+510T2−8p3T3+p6T4 1 - 8 T + 510 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+92T+430pT2+92p3T3+p6T4 1 + 92 T + 430 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+100T+5166T2+100p3T3+p6T4 1 + 100 T + 5166 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+92T+5030T2+92p3T3+p6T4 1 + 92 T + 5030 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−4T+11370T2−4p3T3+p6T4 1 - 4 T + 11370 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−8T+8798T2−8p3T3+p6T4 1 - 8 T + 8798 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−84T+45742T2−84p3T3+p6T4 1 - 84 T + 45742 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−384T+84158T2−384p3T3+p6T4 1 - 384 T + 84158 T^{2} - 384 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−172T+99294T2−172p3T3+p6T4 1 - 172 T + 99294 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−300T+157270T2−300p3T3+p6T4 1 - 300 T + 157270 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−300T+180314T2−300p3T3+p6T4 1 - 300 T + 180314 T^{2} - 300 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+16T+114782T2+16p3T3+p6T4 1 + 16 T + 114782 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+12T+288382T2+12p3T3+p6T4 1 + 12 T + 288382 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−644T+462170T2−644p3T3+p6T4 1 - 644 T + 462170 T^{2} - 644 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+292T+240078T2+292p3T3+p6T4 1 + 292 T + 240078 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1+172T+278250T2+172p3T3+p6T4 1 + 172 T + 278250 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−408T+672766T2−408p3T3+p6T4 1 - 408 T + 672766 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−412T+690678T2−412p3T3+p6T4 1 - 412 T + 690678 T^{2} - 412 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−400T+963870T2−400p3T3+p6T4 1 - 400 T + 963870 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+948T+1360138T2+948p3T3+p6T4 1 + 948 T + 1360138 T^{2} + 948 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+572T+845846T2+572p3T3+p6T4 1 + 572 T + 845846 T^{2} + 572 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−2204T+2633478T2−2204p3T3+p6T4 1 - 2204 T + 2633478 T^{2} - 2204 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

−9.662792480496163355446543600710, −9.441459499161729544598997185917, −8.651910157887449987180556197540, −8.376341288831722219521912745059, −7.85479912013808595673996317229, −7.72141285147767170603437662495, −7.20166815985792174207873748113, −6.85663982471742886323434682254, −6.04895556470567799614040457354, −5.98136192580979019012121248397, −5.06944870754815578145752451427, −5.01924924228876027218602333715, −4.58891629868991337732050299903, −4.23099582514183572993488495344, −3.13682051332412651484221418709, −2.70864990701583260923350479167, −2.23537267017876954408340918840, −2.21298656889584361449449615550, −0.910587434348081480585361841018, −0.29964300976567435285467700142, 0.29964300976567435285467700142, 0.910587434348081480585361841018, 2.21298656889584361449449615550, 2.23537267017876954408340918840, 2.70864990701583260923350479167, 3.13682051332412651484221418709, 4.23099582514183572993488495344, 4.58891629868991337732050299903, 5.01924924228876027218602333715, 5.06944870754815578145752451427, 5.98136192580979019012121248397, 6.04895556470567799614040457354, 6.85663982471742886323434682254, 7.20166815985792174207873748113, 7.72141285147767170603437662495, 7.85479912013808595673996317229, 8.376341288831722219521912745059, 8.651910157887449987180556197540, 9.441459499161729544598997185917, 9.662792480496163355446543600710

Graph of the ZZ-function along the critical line