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  − 50·5-s − 446·9-s + 292·13-s − 1.40e3·17-s + 1.87e3·25-s + 8.02e3·29-s + 2.95e4·37-s − 8.70e3·41-s + 2.23e4·45-s − 3.16e4·49-s + 3.63e4·53-s + 8.42e4·61-s − 1.46e4·65-s + 5.25e4·73-s + 1.39e5·81-s + 7.02e4·85-s + 6.11e4·89-s + 1.33e5·97-s − 8.55e4·101-s + 2.23e5·109-s + 4.33e5·113-s − 1.30e5·117-s + 1.97e5·121-s − 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.83·9-s + 0.479·13-s − 1.17·17-s + 3/5·25-s + 1.77·29-s + 3.54·37-s − 0.808·41-s + 1.64·45-s − 1.88·49-s + 1.77·53-s + 2.89·61-s − 0.428·65-s + 1.15·73-s + 2.36·81-s + 1.05·85-s + 0.818·89-s + 1.44·97-s − 0.834·101-s + 1.80·109-s + 3.19·113-s − 0.879·117-s + 1.22·121-s − 0.357·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-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  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(2.188912979\)
\(L(\frac12)\)  \(\approx\)  \(2.188912979\)
\(L(\frac{7}{2})\)   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$C_1$ \( ( 1 + p^{2} T )^{2} \)
good3$C_2^2$ \( 1 + 446 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 + 646 p^{2} T^{2} + p^{10} T^{4} \)
11$C_2^2$ \( 1 - 197738 T^{2} + p^{10} T^{4} \)
13$C_2$ \( ( 1 - 146 T + p^{5} T^{2} )^{2} \)
17$C_2$ \( ( 1 + 702 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 2512762 T^{2} + p^{10} T^{4} \)
23$C_2^2$ \( 1 - 3871674 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 4010 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 36406942 T^{2} + p^{10} T^{4} \)
37$C_2$ \( ( 1 - 14778 T + p^{5} T^{2} )^{2} \)
41$C_2$ \( ( 1 + 4350 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 139567886 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 + 422513974 T^{2} + p^{10} T^{4} \)
53$C_2$ \( ( 1 - 18154 T + p^{5} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 1041470358 T^{2} + p^{10} T^{4} \)
61$C_2$ \( ( 1 - 42130 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 2438310974 T^{2} + p^{10} T^{4} \)
71$C_2^2$ \( 1 + 1542914862 T^{2} + p^{10} T^{4} \)
73$C_2$ \( ( 1 - 26266 T + p^{5} T^{2} )^{2} \)
79$C_2^2$ \( 1 + 6078817438 T^{2} + p^{10} T^{4} \)
83$C_2^2$ \( 1 - 1875047714 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 30570 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 66882 T + p^{5} T^{2} )^{2} \)
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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

−11.05951685403677456589733495959, −10.93311781962910836716675180448, −9.925657842773283885814753234310, −9.802471878091752943783771775280, −8.839626604553640661833342998693, −8.691431160466668538077911189508, −8.214068219239046015020577711646, −7.955254604777747824361998736997, −7.17746656229355761424452603713, −6.59578816200488806180567876760, −6.16843672185548798440531300352, −5.74475709357530976347433880492, −4.85878873764652386062272315053, −4.60887245719111527880648612409, −3.80814673929387101194792277849, −3.26337482456587039544030833630, −2.61585971034506651945096348300, −2.17113340168535426370999821000, −0.67889234004800761623528849125, −0.64567714477605399038840036874, 0.64567714477605399038840036874, 0.67889234004800761623528849125, 2.17113340168535426370999821000, 2.61585971034506651945096348300, 3.26337482456587039544030833630, 3.80814673929387101194792277849, 4.60887245719111527880648612409, 4.85878873764652386062272315053, 5.74475709357530976347433880492, 6.16843672185548798440531300352, 6.59578816200488806180567876760, 7.17746656229355761424452603713, 7.955254604777747824361998736997, 8.214068219239046015020577711646, 8.691431160466668538077911189508, 8.839626604553640661833342998693, 9.802471878091752943783771775280, 9.925657842773283885814753234310, 10.93311781962910836716675180448, 11.05951685403677456589733495959

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