Properties

Degree 4
Conductor $ 2^{12} \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·5-s + 362·9-s − 200·11-s + 4.48e3·19-s − 3.02e3·25-s − 1.57e4·29-s + 4.28e3·31-s − 1.48e4·41-s + 3.62e3·45-s + 1.86e4·49-s − 2.00e3·55-s − 5.19e4·59-s + 6.11e3·61-s − 7.52e4·71-s + 1.59e5·79-s + 7.19e4·81-s + 1.65e3·89-s + 4.48e4·95-s − 7.24e4·99-s + 2.87e5·101-s − 2.12e5·109-s − 2.92e5·121-s − 6.15e4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.178·5-s + 1.48·9-s − 0.498·11-s + 2.85·19-s − 0.967·25-s − 3.46·29-s + 0.801·31-s − 1.37·41-s + 0.266·45-s + 1.10·49-s − 0.0891·55-s − 1.94·59-s + 0.210·61-s − 1.77·71-s + 2.87·79-s + 1.21·81-s + 0.0221·89-s + 0.510·95-s − 0.742·99-s + 2.80·101-s − 1.71·109-s − 1.81·121-s − 0.352·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(102400\)    =    \(2^{12} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{320} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.723318717\)
\(L(\frac12)\)  \(\approx\)  \(2.723318717\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_2$ \( 1 - 2 p T + p^{5} T^{2} \)
good3$C_2^2$ \( 1 - 362 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 18610 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 + 100 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 202442 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 1879458 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 2244 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1185810 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 7854 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2144 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 30515770 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 7414 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 21442214 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 369731298 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 248174170 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 25972 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3058 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 755362070 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 37608 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3569749522 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 79728 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7612675530 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 826 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 15761405890 T^{2} + p^{10} T^{4} \)
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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

−11.11574585690471616448469564112, −10.36073297503395336830007795927, −10.12581781821141615021545329504, −9.621212577586265929609035630620, −9.271758284613131533267663081524, −8.936388524099825809113371961115, −7.77157959699295499897780676075, −7.66247122572529922476196792655, −7.45625845686601697938979634600, −6.78003575548190241237761128212, −6.10064657929294140039396587696, −5.46181142286028004731511694339, −5.19399462906579573142919958004, −4.52887319864426694364326652379, −3.59536784670161724108992805416, −3.59094956574471590014844025913, −2.56114965143684325321073482418, −1.70459240082522058207036972744, −1.37045245245271877766990722539, −0.44340848046082226308189162315, 0.44340848046082226308189162315, 1.37045245245271877766990722539, 1.70459240082522058207036972744, 2.56114965143684325321073482418, 3.59094956574471590014844025913, 3.59536784670161724108992805416, 4.52887319864426694364326652379, 5.19399462906579573142919958004, 5.46181142286028004731511694339, 6.10064657929294140039396587696, 6.78003575548190241237761128212, 7.45625845686601697938979634600, 7.66247122572529922476196792655, 7.77157959699295499897780676075, 8.936388524099825809113371961115, 9.271758284613131533267663081524, 9.621212577586265929609035630620, 10.12581781821141615021545329504, 10.36073297503395336830007795927, 11.11574585690471616448469564112

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