Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 9-s + 4·13-s + 5·16-s + 6·17-s − 2·18-s − 8·19-s + 25-s − 8·26-s − 6·32-s − 12·34-s + 3·36-s + 16·38-s − 8·43-s + 2·49-s − 2·50-s + 12·52-s − 12·53-s + 7·64-s − 8·67-s + 18·68-s − 4·72-s − 24·76-s + 81-s + 24·83-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1/3·9-s + 1.10·13-s + 5/4·16-s + 1.45·17-s − 0.471·18-s − 1.83·19-s + 1/5·25-s − 1.56·26-s − 1.06·32-s − 2.05·34-s + 1/2·36-s + 2.59·38-s − 1.21·43-s + 2/7·49-s − 0.282·50-s + 1.66·52-s − 1.64·53-s + 7/8·64-s − 0.977·67-s + 2.18·68-s − 0.471·72-s − 2.75·76-s + 1/9·81-s + 2.63·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{260100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 260100,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9083409572$
$L(\frac12)$  $\approx$  $0.9083409572$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 - 6 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.875429388942647608763233005105, −8.540439056490490985128004896836, −7.941231322257043910223245152113, −7.79711986387068141575976779625, −7.19170400608151927655187626887, −6.42217617652666865799421983967, −6.37150043364000269525301859690, −5.82487279118675041342104526304, −5.01944387438421311944798675589, −4.46472359494342999265261204391, −3.45323495695880355625020856878, −3.36585804145949552210873743484, −2.19891095670466256908754466292, −1.67422833535095353270525529506, −0.74314124481069084021545695805, 0.74314124481069084021545695805, 1.67422833535095353270525529506, 2.19891095670466256908754466292, 3.36585804145949552210873743484, 3.45323495695880355625020856878, 4.46472359494342999265261204391, 5.01944387438421311944798675589, 5.82487279118675041342104526304, 6.37150043364000269525301859690, 6.42217617652666865799421983967, 7.19170400608151927655187626887, 7.79711986387068141575976779625, 7.941231322257043910223245152113, 8.540439056490490985128004896836, 8.875429388942647608763233005105

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