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

\(L(1)\) \(\approx\) \(2.986113318\)
\(L(\frac12)\) \(\approx\) \(2.986113318\)
\(L(\frac{3}{2})\) 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

−9.894861910460469390555412424077, −9.888813602463832206969879485326, −9.181383348129956121802864640353, −9.105066421638555384748676992002, −8.779248243319662642077109801314, −8.470803708734452887368954340046, −7.21361517655575084220365219986, −7.18366102472951024186826040104, −6.48071264477018863662709027190, −6.38573516368759475917566712732, −6.30164219622437506199627978794, −5.62146624535533189997124188178, −4.73582601770195859028382620820, −4.64618146519514869628386735818, −3.77057343027783696062868461053, −3.46328607007613271714874120677, −2.67992477069804157496770614128, −2.47121977856574694614672051821, −1.30092379500974649699960088016, −0.933950935034233252038729125675, 0.933950935034233252038729125675, 1.30092379500974649699960088016, 2.47121977856574694614672051821, 2.67992477069804157496770614128, 3.46328607007613271714874120677, 3.77057343027783696062868461053, 4.64618146519514869628386735818, 4.73582601770195859028382620820, 5.62146624535533189997124188178, 6.30164219622437506199627978794, 6.38573516368759475917566712732, 6.48071264477018863662709027190, 7.18366102472951024186826040104, 7.21361517655575084220365219986, 8.470803708734452887368954340046, 8.779248243319662642077109801314, 9.105066421638555384748676992002, 9.181383348129956121802864640353, 9.888813602463832206969879485326, 9.894861910460469390555412424077

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