Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 4
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 + 2·3-s + 8·4-s − 30·5-s + 8·6-s − 38·7-s + 2·9-s − 120·10-s + 404·11-s + 16·12-s − 198·13-s − 152·14-s − 60·15-s − 64·16-s − 478·17-s + 8·18-s − 240·20-s − 76·21-s + 1.61e3·22-s + 1.08e3·23-s + 275·25-s − 792·26-s + 162·27-s − 304·28-s − 240·30-s − 1.51e3·31-s − 256·32-s + ⋯
L(s)  = 1  + 2-s + 2/9·3-s + 1/2·4-s − 6/5·5-s + 2/9·6-s − 0.775·7-s + 2/81·9-s − 6/5·10-s + 3.33·11-s + 1/9·12-s − 1.17·13-s − 0.775·14-s − 0.266·15-s − 1/4·16-s − 1.65·17-s + 2/81·18-s − 3/5·20-s − 0.172·21-s + 3.33·22-s + 2.04·23-s + 0.439·25-s − 1.17·26-s + 2/9·27-s − 0.387·28-s − 0.266·30-s − 1.57·31-s − 1/4·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  induced by $\chi_{10} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 100,\ (\ :2, 2),\ 1)$
$L(\frac{5}{2})$  $\approx$  $1.48740$
$L(\frac12)$  $\approx$  $1.48740$
$L(3)$   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$C_2$ \( 1 + 6 p T + p^{4} T^{2} \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p^{4} T^{3} + p^{8} T^{4} \)
7$C_2^2$ \( 1 + 38 T + 722 T^{2} + 38 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2$ \( ( 1 - 202 T + p^{4} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 198 T + 19602 T^{2} + 198 p^{4} T^{3} + p^{8} T^{4} \)
17$C_2^2$ \( 1 + 478 T + 114242 T^{2} + 478 p^{4} T^{3} + p^{8} T^{4} \)
19$C_2^2$ \( 1 - 259042 T^{2} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 1082 T + 585362 T^{2} - 1082 p^{4} T^{3} + p^{8} T^{4} \)
29$C_2^2$ \( 1 - 1374562 T^{2} + p^{8} T^{4} \)
31$C_2$ \( ( 1 + 758 T + p^{4} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 282 T + 39762 T^{2} - 282 p^{4} T^{3} + p^{8} T^{4} \)
41$C_2$ \( ( 1 - 1042 T + p^{4} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 1518 T + 1152162 T^{2} + 1518 p^{4} T^{3} + p^{8} T^{4} \)
47$C_2^2$ \( 1 + 918 T + 421362 T^{2} + 918 p^{4} T^{3} + p^{8} T^{4} \)
53$C_2^2$ \( 1 + 3638 T + 6617522 T^{2} + 3638 p^{4} T^{3} + p^{8} T^{4} \)
59$C_2^2$ \( 1 - 3074722 T^{2} + p^{8} T^{4} \)
61$C_2$ \( ( 1 - 2082 T + p^{4} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 10162 T + 51633122 T^{2} - 10162 p^{4} T^{3} + p^{8} T^{4} \)
71$C_2$ \( ( 1 + 3478 T + p^{4} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6958 T + 24206882 T^{2} + 6958 p^{4} T^{3} + p^{8} T^{4} \)
79$C_2^2$ \( 1 - 18917762 T^{2} + p^{8} T^{4} \)
83$C_2^2$ \( 1 - 12162 T + 73957122 T^{2} - 12162 p^{4} T^{3} + p^{8} T^{4} \)
89$C_2^2$ \( 1 - 93222082 T^{2} + p^{8} T^{4} \)
97$C_2^2$ \( 1 - 1122 T + 629442 T^{2} - 1122 p^{4} T^{3} + p^{8} T^{4} \)
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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

−20.23607954311365487738523774895, −20.01403317711729005848201304095, −19.27162279492311257663844232801, −19.20368663275779165457972978862, −17.55502990608387111124039039755, −16.98567584052670781622901093783, −16.26268387309593465847672038068, −15.35825312872810959797410897070, −14.54725242579615035720868542985, −14.50150656506435262624987352586, −13.10819895540757872456905795597, −12.51656295742487448191409429372, −11.61489344213984914249970358162, −11.25713092051544583063154212326, −9.349130222125409586860684392726, −8.963181990116567087645578232943, −7.07316282889632245175698041630, −6.58661284276896890433337774218, −4.48822831160574025841244091289, −3.59263323010505890195366133299, 3.59263323010505890195366133299, 4.48822831160574025841244091289, 6.58661284276896890433337774218, 7.07316282889632245175698041630, 8.963181990116567087645578232943, 9.349130222125409586860684392726, 11.25713092051544583063154212326, 11.61489344213984914249970358162, 12.51656295742487448191409429372, 13.10819895540757872456905795597, 14.50150656506435262624987352586, 14.54725242579615035720868542985, 15.35825312872810959797410897070, 16.26268387309593465847672038068, 16.98567584052670781622901093783, 17.55502990608387111124039039755, 19.20368663275779165457972978862, 19.27162279492311257663844232801, 20.01403317711729005848201304095, 20.23607954311365487738523774895

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