Properties

Label 4-294e2-1.1-c6e2-0-0
Degree $4$
Conductor $86436$
Sign $1$
Analytic cond. $4574.61$
Root an. cond. $8.22410$
Motivic weight $6$
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  − 42·3-s − 32·4-s + 1.03e3·9-s + 1.34e3·12-s + 5.90e3·13-s + 1.02e3·16-s − 1.05e4·19-s + 2.45e3·25-s − 1.28e4·27-s − 4.57e4·31-s − 3.31e4·36-s + 6.81e4·37-s − 2.47e5·39-s − 1.28e4·43-s − 4.30e4·48-s − 1.88e5·52-s + 4.41e5·57-s + 1.25e5·61-s − 3.27e4·64-s + 8.77e5·67-s + 1.46e6·73-s − 1.02e5·75-s + 3.36e5·76-s + 6.81e5·79-s − 2.14e5·81-s + 1.92e6·93-s + 5.62e5·97-s + ⋯
L(s)  = 1  − 1.55·3-s − 1/2·4-s + 1.41·9-s + 7/9·12-s + 2.68·13-s + 1/4·16-s − 1.53·19-s + 0.156·25-s − 0.652·27-s − 1.53·31-s − 0.709·36-s + 1.34·37-s − 4.17·39-s − 0.161·43-s − 0.388·48-s − 1.34·52-s + 2.38·57-s + 0.551·61-s − 1/8·64-s + 2.91·67-s + 3.75·73-s − 0.243·75-s + 0.766·76-s + 1.38·79-s − 0.404·81-s + 2.39·93-s + 0.615·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86436\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4574.61\)
Root analytic conductor: \(8.22410\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 86436,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.940359199\)
\(L(\frac12)\) \(\approx\) \(1.940359199\)
\(L(4)\) 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$C_2$ \( 1 + p^{5} T^{2} \)
3$C_2$ \( 1 + 14 p T + p^{6} T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 98 p^{2} T^{2} + p^{12} T^{4} \)
11$C_2^2$ \( 1 - 3541970 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 - 2950 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 28202690 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 + 5258 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 191004770 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 1184779442 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 + 22898 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 34058 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 9219079010 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 + 6406 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10801249342 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 - 7253988050 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 + 22449655150 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 - 62566 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 438698 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 251546372642 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 - 730510 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 340562 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 407613512306 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 844406214050 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 - 281086 T + p^{6} T^{2} )^{2} \)
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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

−10.92130337978273030819896555053, −10.84877536192690724390954381945, −10.16978520753493068531050105518, −9.678561776165801439622618395360, −8.964912962212970413795789652548, −8.715464779560782180618436974184, −8.122551350169719796180131837777, −7.67832725442048462206046043382, −6.71662029938284765300737833904, −6.32717057836488189052794430638, −6.26312424126891370676891768560, −5.41149896765484460196701906779, −5.18019736433294002643670552510, −4.38123606718352929079478690641, −3.73310989117289178792387954244, −3.63805800542901853404959116647, −2.27982238035223091395180484751, −1.59952143282373551629434012210, −0.69289855887798012309923170563, −0.63697940044966219853086672645, 0.63697940044966219853086672645, 0.69289855887798012309923170563, 1.59952143282373551629434012210, 2.27982238035223091395180484751, 3.63805800542901853404959116647, 3.73310989117289178792387954244, 4.38123606718352929079478690641, 5.18019736433294002643670552510, 5.41149896765484460196701906779, 6.26312424126891370676891768560, 6.32717057836488189052794430638, 6.71662029938284765300737833904, 7.67832725442048462206046043382, 8.122551350169719796180131837777, 8.715464779560782180618436974184, 8.964912962212970413795789652548, 9.678561776165801439622618395360, 10.16978520753493068531050105518, 10.84877536192690724390954381945, 10.92130337978273030819896555053

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