Properties

Label 4-1183e2-1.1-c1e2-0-3
Degree $4$
Conductor $1399489$
Sign $1$
Analytic cond. $89.2326$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·4-s − 7-s + 5·8-s + 3·9-s − 3·11-s − 14-s + 5·16-s − 7·17-s + 3·18-s − 7·19-s − 3·22-s + 6·23-s + 5·25-s − 2·28-s − 10·29-s + 10·32-s − 7·34-s + 6·36-s + 8·37-s − 7·38-s + 4·43-s − 6·44-s + 6·46-s + 7·47-s − 6·49-s + 5·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s − 0.377·7-s + 1.76·8-s + 9-s − 0.904·11-s − 0.267·14-s + 5/4·16-s − 1.69·17-s + 0.707·18-s − 1.60·19-s − 0.639·22-s + 1.25·23-s + 25-s − 0.377·28-s − 1.85·29-s + 1.76·32-s − 1.20·34-s + 36-s + 1.31·37-s − 1.13·38-s + 0.609·43-s − 0.904·44-s + 0.884·46-s + 1.02·47-s − 6/7·49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1399489\)    =    \(7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(89.2326\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1183} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1399489,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.010751511\)
\(L(\frac12)\) \(\approx\) \(4.010751511\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−10.06308564520716104313794530665, −9.649953399698223935463595591909, −9.141428851267841831934196126951, −8.786126568384763260614108733579, −8.245483544734073332030914618777, −7.72937871177289099364753892311, −7.36397037398870767500520263020, −7.08420183805453174878321351656, −6.61256860481273696461975222313, −6.32923209422031849582970738260, −5.83694014615232399469479309593, −4.96905673918061977873504007344, −4.92320163728506987877075345213, −4.35108388799355082980131351121, −3.99054887205728175578328385627, −3.41918387351756557942644917843, −2.55110361751634610703236873684, −2.27110672584278211544112184555, −1.77784434914108759578004034314, −0.75827170547943211199786953275, 0.75827170547943211199786953275, 1.77784434914108759578004034314, 2.27110672584278211544112184555, 2.55110361751634610703236873684, 3.41918387351756557942644917843, 3.99054887205728175578328385627, 4.35108388799355082980131351121, 4.92320163728506987877075345213, 4.96905673918061977873504007344, 5.83694014615232399469479309593, 6.32923209422031849582970738260, 6.61256860481273696461975222313, 7.08420183805453174878321351656, 7.36397037398870767500520263020, 7.72937871177289099364753892311, 8.245483544734073332030914618777, 8.786126568384763260614108733579, 9.141428851267841831934196126951, 9.649953399698223935463595591909, 10.06308564520716104313794530665

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