Properties

Label 4-1008e2-1.1-c5e2-0-8
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 28·5-s + 98·7-s + 596·11-s + 532·13-s + 2.10e3·17-s + 2.74e3·19-s + 3.66e3·23-s − 1.55e3·25-s − 720·29-s − 3.97e3·31-s + 2.74e3·35-s + 6.78e3·37-s − 4.00e3·41-s − 1.94e4·43-s − 2.15e4·47-s + 7.20e3·49-s + 5.75e4·53-s + 1.66e4·55-s − 2.23e4·59-s − 3.87e4·61-s + 1.48e4·65-s − 7.28e3·67-s − 3.93e3·71-s + 1.92e4·73-s + 5.84e4·77-s − 1.56e4·79-s − 2.26e4·83-s + ⋯
L(s)  = 1  + 0.500·5-s + 0.755·7-s + 1.48·11-s + 0.873·13-s + 1.76·17-s + 1.74·19-s + 1.44·23-s − 0.498·25-s − 0.158·29-s − 0.743·31-s + 0.378·35-s + 0.815·37-s − 0.371·41-s − 1.60·43-s − 1.42·47-s + 3/7·49-s + 2.81·53-s + 0.743·55-s − 0.835·59-s − 1.33·61-s + 0.437·65-s − 0.198·67-s − 0.0925·71-s + 0.423·73-s + 1.12·77-s − 0.281·79-s − 0.360·83-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: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(8.199689830\)
\(L(\frac12)\) \(\approx\) \(8.199689830\)
\(L(\frac{7}{2})\) 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$D_{4}$ \( 1 - 28 T + 2342 T^{2} - 28 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 596 T + 209810 T^{2} - 596 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 532 T + 747678 T^{2} - 532 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 2100 T + 3445630 T^{2} - 2100 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 2744 T + 6243606 T^{2} - 2744 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 3660 T + 14411722 T^{2} - 3660 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 720 T + 28281754 T^{2} + 720 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 3976 T + 26474190 T^{2} + 3976 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 6788 T + 69768750 T^{2} - 6788 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 4004 T + 203212622 T^{2} + 4004 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 19480 T + 337403910 T^{2} + 19480 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 21560 T + 472446158 T^{2} + 21560 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 57576 T + 1664335546 T^{2} - 57576 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 22344 T + 1540856326 T^{2} + 22344 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 38724 T + 1471462046 T^{2} + 38724 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 7288 T + 1671047286 T^{2} + 7288 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 3932 T + 3486638858 T^{2} + 3932 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 19292 T + 2681901078 T^{2} - 19292 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 15632 T + 5584565598 T^{2} + 15632 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 22624 T + 7152673286 T^{2} + 22624 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 112812 T + 14123112334 T^{2} - 112812 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 36036 T + 8894455622 T^{2} + 36036 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.384992210335621116757054033785, −9.153394181357466592388688580762, −8.629038325855525632269183536767, −8.229131441260520406316309998579, −7.71207520409230729006815982749, −7.44178310519713606810169755747, −6.84652955595899099619094034149, −6.58968175212780053678313279802, −5.82351673965466592760204874527, −5.70489701448378980555468616391, −5.08013643787039121830022660627, −4.90040981801038448679998021240, −3.89593371039786226479630275929, −3.81822274462650904592590653002, −3.12007537402998256553873533521, −2.83704793396413418114170076076, −1.68178219230447011041899347458, −1.54399269226190990087192844989, −1.09279383298476658082285008478, −0.60129495428075790895057899327, 0.60129495428075790895057899327, 1.09279383298476658082285008478, 1.54399269226190990087192844989, 1.68178219230447011041899347458, 2.83704793396413418114170076076, 3.12007537402998256553873533521, 3.81822274462650904592590653002, 3.89593371039786226479630275929, 4.90040981801038448679998021240, 5.08013643787039121830022660627, 5.70489701448378980555468616391, 5.82351673965466592760204874527, 6.58968175212780053678313279802, 6.84652955595899099619094034149, 7.44178310519713606810169755747, 7.71207520409230729006815982749, 8.229131441260520406316309998579, 8.629038325855525632269183536767, 9.153394181357466592388688580762, 9.384992210335621116757054033785

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