Properties

Degree 4
Conductor $ 2^{4} \cdot 5^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 4·5-s − 16·10-s − 74·13-s − 64·16-s + 198·17-s − 32·20-s − 109·25-s − 296·26-s − 256·32-s + 792·34-s − 182·37-s + 944·41-s − 436·50-s − 592·52-s − 54·53-s − 936·61-s − 512·64-s + 296·65-s + 1.58e3·68-s + 506·73-s − 728·74-s + 256·80-s − 729·81-s + 3.77e3·82-s − 792·85-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.357·5-s − 0.505·10-s − 1.57·13-s − 16-s + 2.82·17-s − 0.357·20-s − 0.871·25-s − 2.23·26-s − 1.41·32-s + 3.99·34-s − 0.808·37-s + 3.59·41-s − 1.23·50-s − 1.57·52-s − 0.139·53-s − 1.96·61-s − 64-s + 0.564·65-s + 2.82·68-s + 0.811·73-s − 1.14·74-s + 0.357·80-s − 81-s + 5.08·82-s − 1.01·85-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(400\)    =    \(2^{4} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 400,\ (\ :3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $1.96950$
$L(\frac12)$  $\approx$  $1.96950$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - p^{2} T + p^{3} T^{2} \)
5$C_2$ \( 1 + 4 T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 + p^{6} T^{4} \)
7$C_2^2$ \( 1 + p^{6} T^{4} \)
11$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 92 T + p^{3} T^{2} ) \)
17$C_2$ \( ( 1 - 104 T + p^{3} T^{2} )( 1 - 94 T + p^{3} T^{2} ) \)
19$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 130 T + p^{3} T^{2} )( 1 + 130 T + p^{3} T^{2} ) \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 214 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \)
41$C_2$ \( ( 1 - 472 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 + p^{6} T^{4} \)
47$C_2^2$ \( 1 + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 518 T + p^{3} T^{2} )( 1 + 572 T + p^{3} T^{2} ) \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 468 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + p^{6} T^{4} \)
71$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 1098 T + p^{3} T^{2} )( 1 + 592 T + p^{3} T^{2} ) \)
79$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1670 T + p^{3} T^{2} )( 1 + 1670 T + p^{3} T^{2} ) \)
97$C_2$ \( ( 1 - 594 T + p^{3} T^{2} )( 1 + 1816 T + p^{3} T^{2} ) \)
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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

−18.14684558954369962380112005112, −17.45546087292726170577777845224, −16.69752237222435590939438327202, −16.17593689634869811986342933635, −15.38727699628562890512432007082, −14.74714739455204021924409479449, −14.27626125064912458928709606258, −13.88166295641359620815192458007, −12.69500147627342730624120166724, −12.34280805707255941107442787719, −11.94950548885687097205042938930, −10.99742868321278542442602360451, −9.946969841461214799388836065548, −9.345497442084030485072288587476, −7.83807535551204722256362042230, −7.36530271200470390972058886539, −5.95265495077236844226719802653, −5.26244609920122227016028362377, −4.12937926458756835599315736874, −2.92881682390944771765176714559, 2.92881682390944771765176714559, 4.12937926458756835599315736874, 5.26244609920122227016028362377, 5.95265495077236844226719802653, 7.36530271200470390972058886539, 7.83807535551204722256362042230, 9.345497442084030485072288587476, 9.946969841461214799388836065548, 10.99742868321278542442602360451, 11.94950548885687097205042938930, 12.34280805707255941107442787719, 12.69500147627342730624120166724, 13.88166295641359620815192458007, 14.27626125064912458928709606258, 14.74714739455204021924409479449, 15.38727699628562890512432007082, 16.17593689634869811986342933635, 16.69752237222435590939438327202, 17.45546087292726170577777845224, 18.14684558954369962380112005112

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