Properties

Label 4-1216e2-1.1-c1e2-0-7
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

\(L(1)\) \(\approx\) \(1.993492983\)
\(L(\frac12)\) \(\approx\) \(1.993492983\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
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

−9.892050146137477846198667778016, −9.648076560967476764093598862214, −9.248943792876180282106526506103, −8.888678574573461179085764885642, −8.204052810644402587360936798804, −7.900225712865470621351691421551, −7.44424171897927928764187474818, −7.05006633339860145427753795746, −6.63394717743279723533903728594, −6.35575204356113544667431827066, −5.65396617700276012431969186488, −5.48198430761843276797236102550, −4.76468475562860726325209260769, −4.29876058044247304345007542289, −3.35407486990990152512571686815, −3.29096627504812227289805055781, −2.78225318144012742940223962659, −2.72847360440711388628494488929, −1.21483248383309752623603914887, −0.66564732616554931686820538104, 0.66564732616554931686820538104, 1.21483248383309752623603914887, 2.72847360440711388628494488929, 2.78225318144012742940223962659, 3.29096627504812227289805055781, 3.35407486990990152512571686815, 4.29876058044247304345007542289, 4.76468475562860726325209260769, 5.48198430761843276797236102550, 5.65396617700276012431969186488, 6.35575204356113544667431827066, 6.63394717743279723533903728594, 7.05006633339860145427753795746, 7.44424171897927928764187474818, 7.900225712865470621351691421551, 8.204052810644402587360936798804, 8.888678574573461179085764885642, 9.248943792876180282106526506103, 9.648076560967476764093598862214, 9.892050146137477846198667778016

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