Properties

Label 4-1008e2-1.1-c5e2-0-15
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  − 82·5-s + 98·7-s + 340·11-s + 910·13-s − 3.21e3·17-s + 674·19-s − 1.10e3·23-s + 1.89e3·25-s − 8.06e3·29-s + 6.21e3·31-s − 8.03e3·35-s − 8.51e3·37-s + 1.30e3·41-s + 1.00e4·43-s − 1.27e4·47-s + 7.20e3·49-s + 1.12e4·53-s − 2.78e4·55-s − 1.20e4·59-s + 1.02e5·61-s − 7.46e4·65-s − 2.41e4·67-s + 8.97e4·71-s − 5.55e4·73-s + 3.33e4·77-s − 4.88e4·79-s + 3.57e4·83-s + ⋯
L(s)  = 1  − 1.46·5-s + 0.755·7-s + 0.847·11-s + 1.49·13-s − 2.69·17-s + 0.428·19-s − 0.435·23-s + 0.607·25-s − 1.78·29-s + 1.16·31-s − 1.10·35-s − 1.02·37-s + 0.121·41-s + 0.825·43-s − 0.841·47-s + 3/7·49-s + 0.548·53-s − 1.24·55-s − 0.449·59-s + 3.53·61-s − 2.19·65-s − 0.656·67-s + 2.11·71-s − 1.22·73-s + 0.640·77-s − 0.880·79-s + 0.570·83-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$D_{4}$ \( 1 + 82 T + 4826 T^{2} + 82 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 340 T + 338582 T^{2} - 340 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 70 p T + 2514 p^{2} T^{2} - 70 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 3216 T + 5412958 T^{2} + 3216 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 674 T + 4367142 T^{2} - 674 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 48 p T + 495170 p T^{2} + 48 p^{6} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 8064 T + 52795702 T^{2} + 8064 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 6212 T + 51691038 T^{2} - 6212 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 8512 T + 104326950 T^{2} + 8512 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 1304 T + 73546526 T^{2} - 1304 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 10004 T + 99339510 T^{2} - 10004 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 12748 T + 323438270 T^{2} + 12748 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 11220 T + 664373806 T^{2} - 11220 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 12018 T - 285266426 T^{2} + 12018 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 102738 T + 4326026138 T^{2} - 102738 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 24136 T + 1542084918 T^{2} + 24136 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 89720 T + 4576356302 T^{2} - 89720 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 55588 T + 3902743302 T^{2} + 55588 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 48824 T + 5430110622 T^{2} + 48824 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 35782 T + 4098945062 T^{2} - 35782 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 18300 T + 3539716918 T^{2} - 18300 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 69984 T + 18325482398 T^{2} + 69984 p^{5} T^{3} + p^{10} T^{4} \)
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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

−8.788583736632121018759224213150, −8.555462264747417879930574420370, −8.337453001800300570429596163727, −7.86306698847164462236556414466, −7.30992995182567865005322322296, −6.99776912218785861485492708788, −6.48140196088266776150910194411, −6.24719227751406570541971007091, −5.48658832419772704847951516642, −5.11106346973371617143128620697, −4.28189411681381673905601221170, −4.27542748156182842835518704918, −3.65515025034331237826678027192, −3.58853481527506547326383037838, −2.44900018590490422955780581196, −2.17011482081177402574975964839, −1.36418078776407011893637310241, −1.00244756956887179034387628849, 0, 0, 1.00244756956887179034387628849, 1.36418078776407011893637310241, 2.17011482081177402574975964839, 2.44900018590490422955780581196, 3.58853481527506547326383037838, 3.65515025034331237826678027192, 4.27542748156182842835518704918, 4.28189411681381673905601221170, 5.11106346973371617143128620697, 5.48658832419772704847951516642, 6.24719227751406570541971007091, 6.48140196088266776150910194411, 6.99776912218785861485492708788, 7.30992995182567865005322322296, 7.86306698847164462236556414466, 8.337453001800300570429596163727, 8.555462264747417879930574420370, 8.788583736632121018759224213150

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