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  + 90·5-s + 90·9-s + 504·11-s − 440·19-s + 4.97e3·25-s + 1.38e4·29-s − 1.35e4·31-s − 396·41-s + 8.10e3·45-s + 3.00e4·49-s + 4.53e4·55-s − 4.93e4·59-s + 1.13e4·61-s − 1.06e5·71-s − 1.03e5·79-s − 5.09e4·81-s − 1.99e4·89-s − 3.96e4·95-s + 4.53e4·99-s + 2.18e5·101-s + 4.20e4·109-s − 1.31e5·121-s + 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.60·5-s + 0.370·9-s + 1.25·11-s − 0.279·19-s + 1.59·25-s + 3.06·29-s − 2.52·31-s − 0.0367·41-s + 0.596·45-s + 1.78·49-s + 2.02·55-s − 1.84·59-s + 0.392·61-s − 2.51·71-s − 1.87·79-s − 0.862·81-s − 0.267·89-s − 0.450·95-s + 0.465·99-s + 2.12·101-s + 0.338·109-s − 0.817·121-s + 0.953·125-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  =  \((4,\ 102400,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(5.206347680\)
\(L(\frac12)\)  \(\approx\)  \(5.206347680\)
\(L(\frac{7}{2})\)   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$C_2$ \( 1 - 18 p T + p^{5} T^{2} \)
good3$C_2$ \( ( 1 - 8 p T + p^{5} T^{2} )( 1 + 8 p T + p^{5} T^{2} ) \)
7$C_2^2$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 252 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 + 220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 + 198 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5698 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 + 53352 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 9990 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
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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

−10.80201467164829489701184853755, −10.40892858770794560736968889890, −10.23730349310697508008336373242, −9.565311661586334235127803908271, −9.229959430750911535329708823254, −8.644970969501270960808861750581, −8.616892640421080972514119794961, −7.42688553001147750088325397244, −7.21659796138642341754144825226, −6.50809377868913142067125706158, −6.19374378876673960503909697346, −5.73446475358589556444028298708, −5.12455940119600115867489365799, −4.46279834610613191267000440772, −4.02121513750717320755211096792, −3.09033000781569792967789973524, −2.59172859862248909777134599662, −1.65943898943051853254038489307, −1.47481374772971351464535761853, −0.60490808402206526809814464569, 0.60490808402206526809814464569, 1.47481374772971351464535761853, 1.65943898943051853254038489307, 2.59172859862248909777134599662, 3.09033000781569792967789973524, 4.02121513750717320755211096792, 4.46279834610613191267000440772, 5.12455940119600115867489365799, 5.73446475358589556444028298708, 6.19374378876673960503909697346, 6.50809377868913142067125706158, 7.21659796138642341754144825226, 7.42688553001147750088325397244, 8.616892640421080972514119794961, 8.644970969501270960808861750581, 9.229959430750911535329708823254, 9.565311661586334235127803908271, 10.23730349310697508008336373242, 10.40892858770794560736968889890, 10.80201467164829489701184853755

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