Properties

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

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 50·5-s − 306·9-s − 308·13-s + 356·17-s + 1.87e3·25-s − 8.22e3·29-s − 1.48e4·37-s + 1.45e4·41-s − 1.53e4·45-s − 1.43e4·49-s − 6.44e4·53-s − 5.35e4·61-s − 1.54e4·65-s − 3.70e4·73-s + 3.45e4·81-s + 1.78e4·85-s − 2.15e5·89-s − 2.17e5·97-s − 1.18e5·101-s + 2.79e5·109-s − 8.60e4·113-s + 9.42e4·117-s − 2.54e5·121-s + 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.25·9-s − 0.505·13-s + 0.298·17-s + 3/5·25-s − 1.81·29-s − 1.78·37-s + 1.35·41-s − 1.12·45-s − 0.856·49-s − 3.15·53-s − 1.84·61-s − 0.452·65-s − 0.814·73-s + 0.585·81-s + 0.267·85-s − 2.87·89-s − 2.34·97-s − 1.15·101-s + 2.25·109-s − 0.634·113-s + 0.636·117-s − 1.58·121-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  =  \(2\)
Selberg data  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   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$C_2^2$ \( 1 + 34 p^{2} T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 + 14394 T^{2} + p^{10} T^{4} \)
11$C_2^2$ \( 1 + 254822 T^{2} + p^{10} T^{4} \)
13$C_2$ \( ( 1 + 154 T + p^{5} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 178 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 4019078 T^{2} + p^{10} T^{4} \)
23$C_2^2$ \( 1 + 5934266 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 4110 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 47289582 T^{2} + p^{10} T^{4} \)
37$C_2$ \( ( 1 + 7442 T + p^{5} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 7270 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 26783614 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 + 403777034 T^{2} + p^{10} T^{4} \)
53$C_2$ \( ( 1 + 32226 T + p^{5} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 270997718 T^{2} + p^{10} T^{4} \)
61$C_2$ \( ( 1 + 26770 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 219594834 T^{2} + p^{10} T^{4} \)
71$C_2^2$ \( 1 + 681226622 T^{2} + p^{10} T^{4} \)
73$C_2$ \( ( 1 + 18534 T + p^{5} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 1369983522 T^{2} + p^{10} T^{4} \)
83$C_2^2$ \( 1 + 1693436786 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 107590 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 108838 T + p^{5} T^{2} )^{2} \)
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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

−10.79416707288769738795811510253, −9.945554062805176147158495009041, −9.458577640981746493848796912294, −9.442682909103091105239284018366, −8.600973770804502783041058379535, −8.400409136272377925883903568184, −7.49448363311508899974509643248, −7.42705642816900083699225981619, −6.43465355029748116204503022177, −6.20121045693285093557064583755, −5.47640448027724189083629960911, −5.32568348494236936101353916098, −4.58849657454498014112286828808, −3.82779672076084275475475817841, −3.00121562687584385147275153389, −2.78081457819218618490337064845, −1.78668976466563719064047722148, −1.43708687291332388069920107582, 0, 0, 1.43708687291332388069920107582, 1.78668976466563719064047722148, 2.78081457819218618490337064845, 3.00121562687584385147275153389, 3.82779672076084275475475817841, 4.58849657454498014112286828808, 5.32568348494236936101353916098, 5.47640448027724189083629960911, 6.20121045693285093557064583755, 6.43465355029748116204503022177, 7.42705642816900083699225981619, 7.49448363311508899974509643248, 8.400409136272377925883903568184, 8.600973770804502783041058379535, 9.442682909103091105239284018366, 9.458577640981746493848796912294, 9.945554062805176147158495009041, 10.79416707288769738795811510253

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