Properties

Label 4-33e2-1.1-c3e2-0-1
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $3.79105$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 6·3-s + 9·4-s − 14·5-s − 6·6-s + 24·7-s + 25·8-s + 27·9-s − 14·10-s − 22·11-s − 54·12-s + 30·13-s + 24·14-s + 84·15-s + 41·16-s + 106·17-s + 27·18-s + 50·19-s − 126·20-s − 144·21-s − 22·22-s + 134·23-s − 150·24-s − 6·25-s + 30·26-s − 108·27-s + 216·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.15·3-s + 9/8·4-s − 1.25·5-s − 0.408·6-s + 1.29·7-s + 1.10·8-s + 9-s − 0.442·10-s − 0.603·11-s − 1.29·12-s + 0.640·13-s + 0.458·14-s + 1.44·15-s + 0.640·16-s + 1.51·17-s + 0.353·18-s + 0.603·19-s − 1.40·20-s − 1.49·21-s − 0.213·22-s + 1.21·23-s − 1.27·24-s − 0.0479·25-s + 0.226·26-s − 0.769·27-s + 1.45·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(3.79105\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1089,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.491170299\)
\(L(\frac12)\) \(\approx\) \(1.491170299\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 + p T )^{2} \)
good2$D_{4}$ \( 1 - T - p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
5$D_{4}$ \( 1 + 14 T + 202 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
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

−16.81831671504779142087347297125, −15.71459721248052431777341977517, −15.43766405448659409016357141234, −15.22139744115513506306848104642, −13.99363146756414607205109330575, −13.65385847386436529128082517717, −12.52751069401484390402409559939, −11.93654204009773310562450625040, −11.69814408747768663976064065374, −10.95980477158803958093093921586, −10.79310108261216811024923539595, −9.918859945941867237824488883979, −8.380222906896667839992074325438, −7.66190969758609841466684102712, −7.38687948636624159398935100409, −6.35523633698523852422200744919, −5.27021583651063597511424464986, −4.73356142042452570333742314381, −3.45814931002554557885865185437, −1.40365882153410304370441844951, 1.40365882153410304370441844951, 3.45814931002554557885865185437, 4.73356142042452570333742314381, 5.27021583651063597511424464986, 6.35523633698523852422200744919, 7.38687948636624159398935100409, 7.66190969758609841466684102712, 8.380222906896667839992074325438, 9.918859945941867237824488883979, 10.79310108261216811024923539595, 10.95980477158803958093093921586, 11.69814408747768663976064065374, 11.93654204009773310562450625040, 12.52751069401484390402409559939, 13.65385847386436529128082517717, 13.99363146756414607205109330575, 15.22139744115513506306848104642, 15.43766405448659409016357141234, 15.71459721248052431777341977517, 16.81831671504779142087347297125

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