Properties

Label 4-490e2-1.1-c1e2-0-18
Degree $4$
Conductor $240100$
Sign $1$
Analytic cond. $15.3089$
Root an. cond. $1.97804$
Motivic weight $1$
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  − 2·2-s − 4·3-s + 3·4-s − 2·5-s + 8·6-s − 4·8-s + 8·9-s + 4·10-s + 4·11-s − 12·12-s + 4·13-s + 8·15-s + 5·16-s − 8·17-s − 16·18-s − 4·19-s − 6·20-s − 8·22-s − 8·23-s + 16·24-s + 3·25-s − 8·26-s − 12·27-s − 4·29-s − 16·30-s − 6·32-s − 16·33-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 3/2·4-s − 0.894·5-s + 3.26·6-s − 1.41·8-s + 8/3·9-s + 1.26·10-s + 1.20·11-s − 3.46·12-s + 1.10·13-s + 2.06·15-s + 5/4·16-s − 1.94·17-s − 3.77·18-s − 0.917·19-s − 1.34·20-s − 1.70·22-s − 1.66·23-s + 3.26·24-s + 3/5·25-s − 1.56·26-s − 2.30·27-s − 0.742·29-s − 2.92·30-s − 1.06·32-s − 2.78·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15.3089\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 240100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 20 T + 216 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 152 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−10.73213762512098074802689265877, −10.58610337582269844470310669795, −9.955701569759992037364841241935, −9.515257255811814933643830066743, −8.762847715714262290796433712972, −8.710215133886423869714400237578, −8.027505829786312290029656815937, −7.62368940817791412410796986024, −6.68130388111122230056375947544, −6.67706129260928183873608072134, −6.24121911688125710111233764299, −6.02993191575522903379874152841, −4.99068908344961615257254006412, −4.72405063829918254237162612210, −3.82915150342024593183245811160, −3.53228460490233816266070308549, −1.95349450908375665927514463418, −1.48713854200108292571192289916, 0, 0, 1.48713854200108292571192289916, 1.95349450908375665927514463418, 3.53228460490233816266070308549, 3.82915150342024593183245811160, 4.72405063829918254237162612210, 4.99068908344961615257254006412, 6.02993191575522903379874152841, 6.24121911688125710111233764299, 6.67706129260928183873608072134, 6.68130388111122230056375947544, 7.62368940817791412410796986024, 8.027505829786312290029656815937, 8.710215133886423869714400237578, 8.762847715714262290796433712972, 9.515257255811814933643830066743, 9.955701569759992037364841241935, 10.58610337582269844470310669795, 10.73213762512098074802689265877

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