Properties

Label 4-2112e2-1.1-c3e2-0-8
Degree $4$
Conductor $4460544$
Sign $1$
Analytic cond. $15528.1$
Root an. cond. $11.1629$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 14·5-s − 24·7-s + 27·9-s − 22·11-s − 30·13-s − 84·15-s + 106·17-s + 50·19-s + 144·21-s − 134·23-s − 6·25-s − 108·27-s + 198·29-s − 360·31-s + 132·33-s − 336·35-s + 328·37-s + 180·39-s − 782·41-s + 386·43-s + 378·45-s − 266·47-s + 134·49-s − 636·51-s + 522·53-s − 308·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.25·5-s − 1.29·7-s + 9-s − 0.603·11-s − 0.640·13-s − 1.44·15-s + 1.51·17-s + 0.603·19-s + 1.49·21-s − 1.21·23-s − 0.0479·25-s − 0.769·27-s + 1.26·29-s − 2.08·31-s + 0.696·33-s − 1.62·35-s + 1.45·37-s + 0.739·39-s − 2.97·41-s + 1.36·43-s + 1.25·45-s − 0.825·47-s + 0.390·49-s − 1.74·51-s + 1.35·53-s − 0.755·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4460544\)    =    \(2^{12} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(15528.1\)
Root analytic conductor: \(11.1629\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4460544,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 + p T )^{2} \)
good5$D_{4}$ \( 1 - 14 T + 202 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 30 T + 4522 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 106 T + 7882 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 50 T + 14246 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 134 T + 26398 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 360 T + 90430 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 328 T + 62630 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 386 T + 179870 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 266 T + 92542 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 522 T + 295162 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 172 T + 175654 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 778 T + 577250 T^{2} - 778 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 776 T + 528582 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 630 T + 744334 T^{2} + 630 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1296 T + 1178926 T^{2} - 1296 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 652 T + 589506 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 324 T + 579670 T^{2} + 324 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 756 T + 1427110 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 452 T + 982470 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.472312525780441757180876742787, −8.205539235379976929871056113425, −7.51164920471636100458862320444, −7.42795288589861653139054629920, −6.74383831625972180987990187596, −6.65724078010004304756091399706, −5.91273849510928908582857892651, −5.86061046059353646431171744937, −5.48618737980421678762834421003, −5.21268579424062017996742294943, −4.64199298428006458455751959310, −4.09059139512440551722830023729, −3.46519715384380952153481499954, −3.22386322073883853001893178370, −2.48643864884181134921350768366, −2.10826661934345558133109046725, −1.43590741674930434303043348238, −0.942119045351771133279059287173, 0, 0, 0.942119045351771133279059287173, 1.43590741674930434303043348238, 2.10826661934345558133109046725, 2.48643864884181134921350768366, 3.22386322073883853001893178370, 3.46519715384380952153481499954, 4.09059139512440551722830023729, 4.64199298428006458455751959310, 5.21268579424062017996742294943, 5.48618737980421678762834421003, 5.86061046059353646431171744937, 5.91273849510928908582857892651, 6.65724078010004304756091399706, 6.74383831625972180987990187596, 7.42795288589861653139054629920, 7.51164920471636100458862320444, 8.205539235379976929871056113425, 8.472312525780441757180876742787

Graph of the $Z$-function along the critical line