Properties

Label 4-1008e2-1.1-c5e2-0-13
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 42·5-s + 98·7-s − 716·11-s − 714·13-s + 1.34e3·17-s + 1.94e3·19-s − 1.79e3·23-s − 102·25-s + 1.20e3·29-s + 6.80e3·31-s − 4.11e3·35-s + 1.46e4·37-s − 7.89e3·41-s − 524·43-s − 1.83e4·47-s + 7.20e3·49-s − 4.51e4·53-s + 3.00e4·55-s + 2.25e4·59-s − 5.28e4·61-s + 2.99e4·65-s − 9.84e3·67-s − 840·71-s − 1.22e5·73-s − 7.01e4·77-s − 3.17e4·79-s + 3.69e4·83-s + ⋯
L(s)  = 1  − 0.751·5-s + 0.755·7-s − 1.78·11-s − 1.17·13-s + 1.12·17-s + 1.23·19-s − 0.706·23-s − 0.0326·25-s + 0.264·29-s + 1.27·31-s − 0.567·35-s + 1.75·37-s − 0.733·41-s − 0.0432·43-s − 1.21·47-s + 3/7·49-s − 2.20·53-s + 1.34·55-s + 0.844·59-s − 1.81·61-s + 0.880·65-s − 0.268·67-s − 0.0197·71-s − 2.68·73-s − 1.34·77-s − 0.571·79-s + 0.589·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good5$D_{4}$ \( 1 + 42 T + 1866 T^{2} + 42 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 716 T + 412438 T^{2} + 716 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 714 T + 814258 T^{2} + 714 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1344 T + 2728510 T^{2} - 1344 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 1946 T + 4229670 T^{2} - 1946 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 1792 T + 11254510 T^{2} + 1792 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 1200 T - 16154090 T^{2} - 1200 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 6804 T + 30441118 T^{2} - 6804 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 14640 T + 187693126 T^{2} - 14640 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 7896 T + 209593854 T^{2} + 7896 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 524 T + 242298998 T^{2} + 524 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 18396 T + 435885630 T^{2} + 18396 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 45132 T + 1149514990 T^{2} + 45132 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 22582 T + 1531622854 T^{2} - 22582 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 52822 T + 2145929546 T^{2} + 52822 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 9848 T + 2714812022 T^{2} + 9848 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 840 T + 427300302 T^{2} + 840 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 122052 T + 7590539974 T^{2} + 122052 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 31704 T + 6042098590 T^{2} + 31704 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 36974 T + 4839605030 T^{2} - 36974 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 210588 T + 21813719542 T^{2} - 210588 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 44240 T + 2438219582 T^{2} + 44240 p^{5} T^{3} + p^{10} T^{4} \)
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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

−8.985047645880803731325828853962, −8.404332234211725080814804577153, −7.975210447021038967037422345016, −7.83241754764777569878581022713, −7.46144238864300555088881737048, −7.31090418798391972475280916887, −6.37740747398350402240220344410, −5.99549121160976253599714928601, −5.53773099293817830516093598625, −4.92979529781255504783900853325, −4.66949229757156607419583284129, −4.51904841942047258760632390658, −3.34277304624709977809589753266, −3.32697978225610252534123577396, −2.62416097158728288387396140608, −2.22882059911775026323801605395, −1.39776229940196712807930714986, −0.946620708796242120117184656174, 0, 0, 0.946620708796242120117184656174, 1.39776229940196712807930714986, 2.22882059911775026323801605395, 2.62416097158728288387396140608, 3.32697978225610252534123577396, 3.34277304624709977809589753266, 4.51904841942047258760632390658, 4.66949229757156607419583284129, 4.92979529781255504783900853325, 5.53773099293817830516093598625, 5.99549121160976253599714928601, 6.37740747398350402240220344410, 7.31090418798391972475280916887, 7.46144238864300555088881737048, 7.83241754764777569878581022713, 7.975210447021038967037422345016, 8.404332234211725080814804577153, 8.985047645880803731325828853962

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