Properties

Label 4-1008e2-1.1-c1e2-0-47
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $64.7851$
Root an. cond. $2.83706$
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  + 4·7-s − 4·19-s − 2·25-s − 12·29-s + 16·31-s − 4·37-s + 9·49-s − 12·53-s + 12·59-s − 12·83-s − 8·103-s + 28·109-s + 36·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s − 8·175-s + ⋯
L(s)  = 1  + 1.51·7-s − 0.917·19-s − 2/5·25-s − 2.22·29-s + 2.87·31-s − 0.657·37-s + 9/7·49-s − 1.64·53-s + 1.56·59-s − 1.31·83-s − 0.788·103-s + 2.68·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.07·169-s + 0.0760·173-s − 0.604·175-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(64.7851\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.299663072\)
\(L(\frac12)\) \(\approx\) \(2.299663072\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.17717654675375195319275814981, −9.676006291914819441269344034236, −9.538757810048670710223571516680, −8.601687696210814962771173394443, −8.443688551158381539660317915026, −8.380163627580030627585190924310, −7.51192819520089582143565234675, −7.47724639276814183680221468456, −6.87160452710330688023772681563, −6.24622688126620880327132712726, −5.91276184414972441064452187670, −5.43450964107125108163893723856, −4.74257174425157228390661136450, −4.62942892014019535409008522987, −4.05998227547414099170188174269, −3.45392698246985011461660649199, −2.78389696342066649299941617996, −1.97226932497993461772976755695, −1.76129286182797138282354903230, −0.70507083385500222514702759654, 0.70507083385500222514702759654, 1.76129286182797138282354903230, 1.97226932497993461772976755695, 2.78389696342066649299941617996, 3.45392698246985011461660649199, 4.05998227547414099170188174269, 4.62942892014019535409008522987, 4.74257174425157228390661136450, 5.43450964107125108163893723856, 5.91276184414972441064452187670, 6.24622688126620880327132712726, 6.87160452710330688023772681563, 7.47724639276814183680221468456, 7.51192819520089582143565234675, 8.380163627580030627585190924310, 8.443688551158381539660317915026, 8.601687696210814962771173394443, 9.538757810048670710223571516680, 9.676006291914819441269344034236, 10.17717654675375195319275814981

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