Properties

Degree 4
Conductor $ 5^{2} \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s − 4-s − 2·5-s + 4·6-s − 8·8-s + 2·9-s − 4·10-s − 2·11-s − 2·12-s + 6·13-s − 4·15-s − 7·16-s + 2·17-s + 4·18-s − 10·19-s + 2·20-s − 4·22-s + 6·23-s − 16·24-s − 25-s + 12·26-s + 6·27-s − 8·30-s + 10·31-s + 14·32-s − 4·33-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.63·6-s − 2.82·8-s + 2/3·9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s + 1.66·13-s − 1.03·15-s − 7/4·16-s + 0.485·17-s + 0.942·18-s − 2.29·19-s + 0.447·20-s − 0.852·22-s + 1.25·23-s − 3.26·24-s − 1/5·25-s + 2.35·26-s + 1.15·27-s − 1.46·30-s + 1.79·31-s + 2.47·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4225\)    =    \(5^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{65} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4225,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.35216$
$L(\frac12)$  $\approx$  $1.35216$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2$ \( 1 + 2 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−15.14223922415733873819639039462, −14.57493038135502634456586473826, −13.87875728445781756249227402376, −13.52274998356247544693642718630, −13.31812493799828575988047982210, −12.65265466309089736042431370074, −12.16067230501220492707175245836, −11.62722793832416919113155724919, −10.54005650859245523922722823536, −10.24348985741680663624456227474, −8.853818757476344028243810099807, −8.841673258484196603041226504005, −8.436669596736453868347650327525, −7.66232163139685793882688117314, −6.41972131090882448749640525167, −5.93277188311989188901804805163, −4.62117467209420483661964835873, −4.46410220828931050158684313703, −3.44443955823377383091885803600, −3.03470510595937031587764894302, 3.03470510595937031587764894302, 3.44443955823377383091885803600, 4.46410220828931050158684313703, 4.62117467209420483661964835873, 5.93277188311989188901804805163, 6.41972131090882448749640525167, 7.66232163139685793882688117314, 8.436669596736453868347650327525, 8.841673258484196603041226504005, 8.853818757476344028243810099807, 10.24348985741680663624456227474, 10.54005650859245523922722823536, 11.62722793832416919113155724919, 12.16067230501220492707175245836, 12.65265466309089736042431370074, 13.31812493799828575988047982210, 13.52274998356247544693642718630, 13.87875728445781756249227402376, 14.57493038135502634456586473826, 15.14223922415733873819639039462

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