Properties

Label 4-288e2-1.1-c2e2-0-0
Degree $4$
Conductor $82944$
Sign $1$
Analytic cond. $61.5821$
Root an. cond. $2.80132$
Motivic weight $2$
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  − 16·7-s − 16·13-s − 64·19-s + 48·25-s − 80·31-s − 52·37-s − 32·43-s + 94·49-s − 108·61-s + 160·67-s + 192·73-s + 208·79-s + 256·91-s − 160·97-s + 144·103-s − 176·109-s + 114·121-s + 127-s + 131-s + 1.02e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.28·7-s − 1.23·13-s − 3.36·19-s + 1.91·25-s − 2.58·31-s − 1.40·37-s − 0.744·43-s + 1.91·49-s − 1.77·61-s + 2.38·67-s + 2.63·73-s + 2.63·79-s + 2.81·91-s − 1.64·97-s + 1.39·103-s − 1.61·109-s + 0.942·121-s + 0.00787·127-s + 0.00763·131-s + 7.69·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82944\)    =    \(2^{10} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(61.5821\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{288} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82944,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1894753793\)
\(L(\frac12)\) \(\approx\) \(0.1894753793\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 48 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 114 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 416 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 32 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 94 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 + 240 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 1056 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4290 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4560 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 6450 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 54 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 80 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 3810 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 96 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 104 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 3410 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 9792 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 80 T + p^{2} T^{2} )^{2} \)
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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

−12.22857003373973740171699487896, −10.94757156294463995196941838840, −10.94164893526854888733324597848, −10.47381450550739398088698647839, −9.920301999291649889079970520926, −9.342249220004552756002636462748, −9.217901476143042461014860598816, −8.532870871480010398453699439138, −8.129723470136500044750286897979, −7.02990048513921316717427210378, −7.00320858422775567712672509636, −6.42665590160090307109560207031, −6.12922508613007863462554769459, −5.14266872135430825469291939399, −4.78758851543383540519345762828, −3.65477442109780863928056129087, −3.60929617008326486415772068251, −2.56393577355768179018727537940, −2.05199661254050362791877895228, −0.19954119930110532575063739787, 0.19954119930110532575063739787, 2.05199661254050362791877895228, 2.56393577355768179018727537940, 3.60929617008326486415772068251, 3.65477442109780863928056129087, 4.78758851543383540519345762828, 5.14266872135430825469291939399, 6.12922508613007863462554769459, 6.42665590160090307109560207031, 7.00320858422775567712672509636, 7.02990048513921316717427210378, 8.129723470136500044750286897979, 8.532870871480010398453699439138, 9.217901476143042461014860598816, 9.342249220004552756002636462748, 9.920301999291649889079970520926, 10.47381450550739398088698647839, 10.94164893526854888733324597848, 10.94757156294463995196941838840, 12.22857003373973740171699487896

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