Properties

Label 4-2736e2-1.1-c1e2-0-27
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $477.294$
Root an. cond. $4.67408$
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  + 2·5-s − 6·7-s + 4·11-s + 13-s + 6·17-s + 8·19-s + 4·23-s + 5·25-s − 2·29-s − 14·31-s − 12·35-s + 2·37-s − 8·41-s + 7·43-s − 8·47-s + 13·49-s + 8·53-s + 8·55-s + 12·59-s − 5·61-s + 2·65-s − 9·67-s − 2·71-s + 15·73-s − 24·77-s + 11·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.894·5-s − 2.26·7-s + 1.20·11-s + 0.277·13-s + 1.45·17-s + 1.83·19-s + 0.834·23-s + 25-s − 0.371·29-s − 2.51·31-s − 2.02·35-s + 0.328·37-s − 1.24·41-s + 1.06·43-s − 1.16·47-s + 13/7·49-s + 1.09·53-s + 1.07·55-s + 1.56·59-s − 0.640·61-s + 0.248·65-s − 1.09·67-s − 0.237·71-s + 1.75·73-s − 2.73·77-s + 1.23·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(477.294\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.681102391\)
\(L(\frac12)\) \(\approx\) \(2.681102391\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−9.125290788127427790816947691961, −9.048010214949952576064860281132, −8.278222918047085923496680445435, −7.80638534250552728890218063465, −7.31295227896372050019358837635, −7.09087660869865506366533887771, −6.65740399856390852467740553797, −6.37842240954551453391717228906, −6.04027037726084458935113279692, −5.43230550330623335016838517610, −5.37682671200619408122051014423, −4.95589979745193449120180326619, −3.95328821760891640198707889357, −3.66211487330737531111729737125, −3.37784190423396212618533413164, −3.07644033522704432918393632944, −2.50536182422587660507874318240, −1.71234140565390077599417743962, −1.20703424340585459349894803687, −0.58593917508299354324145172196, 0.58593917508299354324145172196, 1.20703424340585459349894803687, 1.71234140565390077599417743962, 2.50536182422587660507874318240, 3.07644033522704432918393632944, 3.37784190423396212618533413164, 3.66211487330737531111729737125, 3.95328821760891640198707889357, 4.95589979745193449120180326619, 5.37682671200619408122051014423, 5.43230550330623335016838517610, 6.04027037726084458935113279692, 6.37842240954551453391717228906, 6.65740399856390852467740553797, 7.09087660869865506366533887771, 7.31295227896372050019358837635, 7.80638534250552728890218063465, 8.278222918047085923496680445435, 9.048010214949952576064860281132, 9.125290788127427790816947691961

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