Properties

Label 4-547e2-1.1-c1e2-0-0
Degree $4$
Conductor $299209$
Sign $1$
Analytic cond. $19.0778$
Root an. cond. $2.08993$
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-s + 4·7-s − 4·9-s − 10·11-s + 6·13-s − 8·14-s + 16-s − 12·17-s + 8·18-s − 2·19-s + 20·22-s − 4·23-s − 8·25-s − 12·26-s + 4·28-s − 14·29-s + 4·31-s + 2·32-s + 24·34-s − 4·36-s − 4·37-s + 4·38-s + 8·41-s − 16·43-s − 10·44-s + 8·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.51·7-s − 4/3·9-s − 3.01·11-s + 1.66·13-s − 2.13·14-s + 1/4·16-s − 2.91·17-s + 1.88·18-s − 0.458·19-s + 4.26·22-s − 0.834·23-s − 8/5·25-s − 2.35·26-s + 0.755·28-s − 2.59·29-s + 0.718·31-s + 0.353·32-s + 4.11·34-s − 2/3·36-s − 0.657·37-s + 0.648·38-s + 1.24·41-s − 2.43·43-s − 1.50·44-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(299209\)    =    \(547^{2}\)
Sign: $1$
Analytic conductor: \(19.0778\)
Root analytic conductor: \(2.08993\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 299209,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad547$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 99 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 22 T + 213 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 140 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 115 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 124 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 180 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 18 T + 203 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−10.49727708143930828980924882194, −10.40063010871114336976822735121, −9.512022582310459563424616343479, −9.076264519747116055547944818013, −8.599339862582055869684959047656, −8.596189569536951409214090985081, −7.916683955956180238956540162057, −7.81961509469193250663324720191, −7.45993975214094272726387586835, −6.27310751784131498687832677238, −6.06739624303554491642745927235, −5.57940107109254159457101416153, −4.92761303449621945411961259473, −4.55628896775451294362478734772, −3.78761205218679729601073493312, −2.92500660130848787551847651303, −2.06133816657720579936252610475, −1.97553677381187046264929544760, 0, 0, 1.97553677381187046264929544760, 2.06133816657720579936252610475, 2.92500660130848787551847651303, 3.78761205218679729601073493312, 4.55628896775451294362478734772, 4.92761303449621945411961259473, 5.57940107109254159457101416153, 6.06739624303554491642745927235, 6.27310751784131498687832677238, 7.45993975214094272726387586835, 7.81961509469193250663324720191, 7.916683955956180238956540162057, 8.596189569536951409214090985081, 8.599339862582055869684959047656, 9.076264519747116055547944818013, 9.512022582310459563424616343479, 10.40063010871114336976822735121, 10.49727708143930828980924882194

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