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 − 4·7-s − 3·11-s + 9·17-s + 7·19-s + 15·23-s + 25-s + 12·29-s + 5·31-s + 12·35-s + 5·37-s − 3·47-s + 9·49-s − 9·53-s + 9·55-s − 9·59-s + 15·61-s − 9·67-s + 3·73-s + 12·77-s + 9·79-s + 24·83-s − 27·85-s + 21·89-s − 21·95-s + 9·101-s − 103-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.51·7-s − 0.904·11-s + 2.18·17-s + 1.60·19-s + 3.12·23-s + 1/5·25-s + 2.22·29-s + 0.898·31-s + 2.02·35-s + 0.821·37-s − 0.437·47-s + 9/7·49-s − 1.23·53-s + 1.21·55-s − 1.17·59-s + 1.92·61-s − 1.09·67-s + 0.351·73-s + 1.36·77-s + 1.01·79-s + 2.63·83-s − 2.92·85-s + 2.22·89-s − 2.15·95-s + 0.895·101-s − 0.0985·103-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

\(L(1)\) \(\approx\) \(1.556310405\)
\(L(\frac12)\) \(\approx\) \(1.556310405\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 15 T + 98 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 21 T + 236 T^{2} - 21 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−10.20772464581508182193139810827, −9.678633000191916942296011043060, −9.311231069464597711469604589178, −9.196984923626832527721402007778, −8.253093769894076859742591889342, −8.079951732229647135050464620021, −7.66668604542390609621200542253, −7.38804932730689304189785873614, −6.66638701623869078768142136065, −6.64567947275818139807415883882, −5.86876215308068170833862885488, −5.37141986948431102430541794778, −4.84179636730765277610529246676, −4.67308370474788940286620195938, −3.55597129459332455583221588646, −3.41647627513991490218230436430, −2.98321202324996564331742231004, −2.68758930852754300653482093215, −1.01797583092597064386126394471, −0.76715783948987930153868730182, 0.76715783948987930153868730182, 1.01797583092597064386126394471, 2.68758930852754300653482093215, 2.98321202324996564331742231004, 3.41647627513991490218230436430, 3.55597129459332455583221588646, 4.67308370474788940286620195938, 4.84179636730765277610529246676, 5.37141986948431102430541794778, 5.86876215308068170833862885488, 6.64567947275818139807415883882, 6.66638701623869078768142136065, 7.38804932730689304189785873614, 7.66668604542390609621200542253, 8.079951732229647135050464620021, 8.253093769894076859742591889342, 9.196984923626832527721402007778, 9.311231069464597711469604589178, 9.678633000191916942296011043060, 10.20772464581508182193139810827

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