Properties

Degree 4
Conductor $ 2^{8} \cdot 503^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 4·7-s − 9-s + 4·11-s − 2·13-s + 4·17-s + 2·19-s − 2·25-s − 10·29-s − 2·31-s − 8·35-s − 6·41-s − 16·43-s + 2·45-s − 4·47-s + 3·49-s + 8·53-s − 8·55-s + 6·61-s − 4·63-s + 4·65-s + 4·67-s − 6·71-s − 24·73-s + 16·77-s − 8·79-s − 8·81-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.970·17-s + 0.458·19-s − 2/5·25-s − 1.85·29-s − 0.359·31-s − 1.35·35-s − 0.937·41-s − 2.43·43-s + 0.298·45-s − 0.583·47-s + 3/7·49-s + 1.09·53-s − 1.07·55-s + 0.768·61-s − 0.503·63-s + 0.496·65-s + 0.488·67-s − 0.712·71-s − 2.80·73-s + 1.82·77-s − 0.900·79-s − 8/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 64770304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 64770304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(64770304\)    =    \(2^{8} \cdot 503^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{8048} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 64770304,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;503\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;503\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2 \( 1 \)
503$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 16 T + 145 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 20 T + 221 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.70853080026506351883304197473, −7.25445600366208856916394748756, −7.02969309791998238552444737320, −6.91832467020040870539121620436, −6.26966062804210803846889956209, −5.75961316903722887355739237620, −5.46276534061150849656010852242, −5.29974653453893358643986619060, −4.92347604667195376247806181486, −4.19767104223064373053467955791, −4.19715501768570964061798392760, −3.89420992044347418125940503392, −3.19262924857777904944509575836, −3.12137151109480175688563203292, −2.46492246749014708356224547771, −1.72915620013153467827217334746, −1.44469799375910412984329104637, −1.34075904265623587125103303333, 0, 0, 1.34075904265623587125103303333, 1.44469799375910412984329104637, 1.72915620013153467827217334746, 2.46492246749014708356224547771, 3.12137151109480175688563203292, 3.19262924857777904944509575836, 3.89420992044347418125940503392, 4.19715501768570964061798392760, 4.19767104223064373053467955791, 4.92347604667195376247806181486, 5.29974653453893358643986619060, 5.46276534061150849656010852242, 5.75961316903722887355739237620, 6.26966062804210803846889956209, 6.91832467020040870539121620436, 7.02969309791998238552444737320, 7.25445600366208856916394748756, 7.70853080026506351883304197473

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