Properties

Label 4-539e2-1.1-c1e2-0-6
Degree $4$
Conductor $290521$
Sign $1$
Analytic cond. $18.5238$
Root an. cond. $2.07459$
Motivic weight $1$
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  + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 4·8-s + 3·9-s + 2·10-s − 11-s − 2·12-s − 8·13-s − 15-s + 8·16-s − 2·17-s + 6·18-s + 2·20-s − 2·22-s + 23-s − 4·24-s + 5·25-s − 16·26-s − 8·27-s − 2·30-s + 7·31-s + 8·32-s + 33-s − 4·34-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.41·8-s + 9-s + 0.632·10-s − 0.301·11-s − 0.577·12-s − 2.21·13-s − 0.258·15-s + 2·16-s − 0.485·17-s + 1.41·18-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 0.816·24-s + 25-s − 3.13·26-s − 1.53·27-s − 0.365·30-s + 1.25·31-s + 1.41·32-s + 0.174·33-s − 0.685·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(290521\)    =    \(7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(18.5238\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 290521,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.648250151\)
\(L(\frac12)\) \(\approx\) \(3.648250151\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
11$C_2$ \( 1 + T + T^{2} \)
good2$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 12 T + 83 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−11.06721126792996225281890271758, −10.58260609971970333494181149261, −10.32467625614385717755264612114, −9.824064317429458425643783876155, −9.571415605014629107687188324938, −8.931063904650405592360426580297, −8.162741939147974914382345281614, −7.56645059217730979516684969241, −7.42275819568418729090796066271, −6.71699091693613544161400637970, −6.60237174557722222786844192066, −5.63578811875978607975840195731, −5.39455014640119999984840278926, −4.82097633270886272408152654355, −4.63122778170841813451460445300, −4.07236695748109229217440871245, −3.41957232484765275166904434092, −2.32601443659742790056971909499, −2.27302924596790538133867218119, −0.968236289550503584798729107264, 0.968236289550503584798729107264, 2.27302924596790538133867218119, 2.32601443659742790056971909499, 3.41957232484765275166904434092, 4.07236695748109229217440871245, 4.63122778170841813451460445300, 4.82097633270886272408152654355, 5.39455014640119999984840278926, 5.63578811875978607975840195731, 6.60237174557722222786844192066, 6.71699091693613544161400637970, 7.42275819568418729090796066271, 7.56645059217730979516684969241, 8.162741939147974914382345281614, 8.931063904650405592360426580297, 9.571415605014629107687188324938, 9.824064317429458425643783876155, 10.32467625614385717755264612114, 10.58260609971970333494181149261, 11.06721126792996225281890271758

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