Properties

Label 4-1470e2-1.1-c3e2-0-4
Degree $4$
Conductor $2160900$
Sign $1$
Analytic cond. $7522.57$
Root an. cond. $9.31304$
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  + 4·2-s − 6·3-s + 12·4-s − 10·5-s − 24·6-s + 32·8-s + 27·9-s − 40·10-s + 2·11-s − 72·12-s − 28·13-s + 60·15-s + 80·16-s − 14·17-s + 108·18-s − 42·19-s − 120·20-s + 8·22-s + 50·23-s − 192·24-s + 75·25-s − 112·26-s − 108·27-s + 78·29-s + 240·30-s − 128·31-s + 192·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s + 0.0548·11-s − 1.73·12-s − 0.597·13-s + 1.03·15-s + 5/4·16-s − 0.199·17-s + 1.41·18-s − 0.507·19-s − 1.34·20-s + 0.0775·22-s + 0.453·23-s − 1.63·24-s + 3/5·25-s − 0.844·26-s − 0.769·27-s + 0.499·29-s + 1.46·30-s − 0.741·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7522.57\)
Root analytic conductor: \(9.31304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.062148387\)
\(L(\frac12)\) \(\approx\) \(5.062148387\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3$C_1$ \( ( 1 + p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 2 T - 778 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 14 T + p^{3} T^{2} )^{2} \)
17$D_{4}$ \( 1 + 14 T + 6434 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 42 T + 10718 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 50 T + 21518 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 78 T + 46858 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 128 T + 49914 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 238 T + 84498 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 292 T + 104102 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 278 T + 92310 T^{2} - 278 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 62 T + 205166 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 226 T + 5794 p T^{2} - 226 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 76 T + 288326 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 242 T + 189882 T^{2} - 242 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 82 T + 434598 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 744 T + 799150 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 320 T + 459534 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1400 T + 1421022 T^{2} - 1400 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 268 T + 562 p T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1660 T + 2043782 T^{2} - 1660 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 264 T + 1498670 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.178256931858085382770078519730, −9.178108297476811219908477872831, −8.285028067689072614143300920518, −8.010325515827850287197450825367, −7.50874841835553970637284646818, −7.17590711009342092650578048075, −6.64055242566244693007848899662, −6.60408452525763754501070068020, −5.89396890174536502853715238159, −5.61509151825348416132159340240, −5.06626008105120193729202689367, −4.79989299070606006997610333007, −4.18971298356845372874590415576, −4.18417181301425286585875159145, −3.30705283411979056174007514975, −3.12909961375530814675759574013, −2.14379822567965332446502669125, −1.95256515460251027586205050747, −0.76725256969207137254230951159, −0.59466088595856532384070395081, 0.59466088595856532384070395081, 0.76725256969207137254230951159, 1.95256515460251027586205050747, 2.14379822567965332446502669125, 3.12909961375530814675759574013, 3.30705283411979056174007514975, 4.18417181301425286585875159145, 4.18971298356845372874590415576, 4.79989299070606006997610333007, 5.06626008105120193729202689367, 5.61509151825348416132159340240, 5.89396890174536502853715238159, 6.60408452525763754501070068020, 6.64055242566244693007848899662, 7.17590711009342092650578048075, 7.50874841835553970637284646818, 8.010325515827850287197450825367, 8.285028067689072614143300920518, 9.178108297476811219908477872831, 9.178256931858085382770078519730

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