Properties

Label 4-1216e2-1.1-c1e2-0-21
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  − 3-s + 8·7-s + 3·9-s + 6·11-s + 2·13-s + 6·17-s − 7·19-s − 8·21-s − 6·23-s + 5·25-s − 8·27-s − 4·31-s − 6·33-s + 20·37-s − 2·39-s − 9·41-s + 4·43-s + 34·49-s − 6·51-s + 6·53-s + 7·57-s + 9·59-s − 4·61-s + 24·63-s + 7·67-s + 6·69-s − 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 3.02·7-s + 9-s + 1.80·11-s + 0.554·13-s + 1.45·17-s − 1.60·19-s − 1.74·21-s − 1.25·23-s + 25-s − 1.53·27-s − 0.718·31-s − 1.04·33-s + 3.28·37-s − 0.320·39-s − 1.40·41-s + 0.609·43-s + 34/7·49-s − 0.840·51-s + 0.824·53-s + 0.927·57-s + 1.17·59-s − 0.512·61-s + 3.02·63-s + 0.855·67-s + 0.722·69-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.843383736\)
\(L(\frac12)\) \(\approx\) \(3.843383736\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−9.816205521790821191649931447627, −9.702053998763435779040645437067, −9.031136496353955283587810119219, −8.594997238356258866197640668077, −8.195864169652111299420725846120, −8.056546432644368456704969429313, −7.42396357932175895233553649356, −7.25598324195587839297105434044, −6.50431025541065414667769076244, −6.18713490506422679275455789007, −5.65996942841242728050958522988, −5.32444225123236410546874657033, −4.67106524340031421225849471868, −4.35404747492204119635745182003, −4.02437107510888914224771831066, −3.69415207664286162707402443037, −2.42178249573288638223281124557, −1.90330062606137141653776220920, −1.26275722374959199203118865377, −1.14937865787251784627735637386, 1.14937865787251784627735637386, 1.26275722374959199203118865377, 1.90330062606137141653776220920, 2.42178249573288638223281124557, 3.69415207664286162707402443037, 4.02437107510888914224771831066, 4.35404747492204119635745182003, 4.67106524340031421225849471868, 5.32444225123236410546874657033, 5.65996942841242728050958522988, 6.18713490506422679275455789007, 6.50431025541065414667769076244, 7.25598324195587839297105434044, 7.42396357932175895233553649356, 8.056546432644368456704969429313, 8.195864169652111299420725846120, 8.594997238356258866197640668077, 9.031136496353955283587810119219, 9.702053998763435779040645437067, 9.816205521790821191649931447627

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