Properties

Label 4-363e2-1.1-c3e2-0-4
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $458.717$
Root an. cond. $4.62792$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

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 − 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 + 134·23-s + 150·24-s − 6·25-s + 30·26-s − 108·27-s − 216·28-s + 198·29-s − 84·30-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 − 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 + 1.21·23-s + 1.27·24-s − 0.0479·25-s + 0.226·26-s − 0.769·27-s − 1.45·28-s + 1.26·29-s − 0.511·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(458.717\)
Root analytic conductor: \(4.62792\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 131769,\ (\ :3/2, 3/2),\ 1)\)

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)$
bad3$C_1$ \( ( 1 + p T )^{2} \)
11 \( 1 \)
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

−10.81855941749096344668129816140, −10.58295092407650123014605870380, −9.832512218723533259092447069774, −9.612257831573193776172132670074, −8.831444691886818151764838637747, −8.542911508523817285204120787867, −7.81296683884895451903654150434, −7.23369437708283712577341620555, −6.81617544536259023433720957606, −6.59208913294589873558272119908, −6.15240073226701506408318090879, −5.59724417245814308492542955500, −4.54193665197481135251553781018, −4.47923496325627273840145314801, −3.51553720050145931323056031409, −2.82115545269818782078599618056, −2.39876748337441984112988073793, −1.09710358176811817074855568181, 0, 0, 1.09710358176811817074855568181, 2.39876748337441984112988073793, 2.82115545269818782078599618056, 3.51553720050145931323056031409, 4.47923496325627273840145314801, 4.54193665197481135251553781018, 5.59724417245814308492542955500, 6.15240073226701506408318090879, 6.59208913294589873558272119908, 6.81617544536259023433720957606, 7.23369437708283712577341620555, 7.81296683884895451903654150434, 8.542911508523817285204120787867, 8.831444691886818151764838637747, 9.612257831573193776172132670074, 9.832512218723533259092447069774, 10.58295092407650123014605870380, 10.81855941749096344668129816140

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