Properties

Degree $4$
Conductor $1016064$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 5·7-s − 9·11-s − 6·17-s − 2·19-s − 12·23-s + 25-s + 18·29-s − 31-s + 15·35-s + 2·37-s + 18·49-s + 9·53-s − 27·55-s − 3·59-s − 12·61-s + 12·73-s − 45·77-s − 3·79-s + 30·83-s − 18·85-s + 18·89-s − 6·95-s + 24·101-s − 4·103-s + 15·107-s + 4·109-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.88·7-s − 2.71·11-s − 1.45·17-s − 0.458·19-s − 2.50·23-s + 1/5·25-s + 3.34·29-s − 0.179·31-s + 2.53·35-s + 0.328·37-s + 18/7·49-s + 1.23·53-s − 3.64·55-s − 0.390·59-s − 1.53·61-s + 1.40·73-s − 5.12·77-s − 0.337·79-s + 3.29·83-s − 1.95·85-s + 1.90·89-s − 0.615·95-s + 2.38·101-s − 0.394·103-s + 1.45·107-s + 0.383·109-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: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.447264544\)
\(L(\frac12)\) \(\approx\) \(2.447264544\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
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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

−10.30137223448153356439117196703, −9.975045519953463060562262920260, −9.392456913647069536027902577335, −8.730587142109839978131006742470, −8.482494123929145211904405253515, −8.100030970195245716863865335897, −7.76943450381297701986946427752, −7.50530572768532093589279947603, −6.68284493660494082509932457183, −6.08832961564354635157744949681, −6.02551987270255980345549577264, −5.36117816755152438890553138862, −4.80945232585049918090712002554, −4.74839696643501832681308898618, −4.25638641070291280378823536336, −3.25317360801401223048450108988, −2.35382533684508749216753455626, −2.20468572041481814113360409137, −1.98003035227879247281892375450, −0.68409286285042670480214725991, 0.68409286285042670480214725991, 1.98003035227879247281892375450, 2.20468572041481814113360409137, 2.35382533684508749216753455626, 3.25317360801401223048450108988, 4.25638641070291280378823536336, 4.74839696643501832681308898618, 4.80945232585049918090712002554, 5.36117816755152438890553138862, 6.02551987270255980345549577264, 6.08832961564354635157744949681, 6.68284493660494082509932457183, 7.50530572768532093589279947603, 7.76943450381297701986946427752, 8.100030970195245716863865335897, 8.482494123929145211904405253515, 8.730587142109839978131006742470, 9.392456913647069536027902577335, 9.975045519953463060562262920260, 10.30137223448153356439117196703

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