Properties

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

Origins

Origins of factors

Downloads

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·8-s + 3·9-s − 8·12-s − 5·13-s + 3·16-s + 3·17-s − 9·18-s − 6·19-s + 6·23-s + 6·24-s + 7·25-s + 15·26-s − 10·27-s − 3·29-s − 6·32-s − 9·34-s + 12·36-s + 15·37-s + 18·38-s + 10·39-s − 9·41-s − 8·43-s − 18·46-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.06·8-s + 9-s − 2.30·12-s − 1.38·13-s + 3/4·16-s + 0.727·17-s − 2.12·18-s − 1.37·19-s + 1.25·23-s + 1.22·24-s + 7/5·25-s + 2.94·26-s − 1.92·27-s − 0.557·29-s − 1.06·32-s − 1.54·34-s + 2·36-s + 2.46·37-s + 2.91·38-s + 1.60·39-s − 1.40·41-s − 1.21·43-s − 2.65·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(169\)    =    \(13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{169} (1, \cdot )$
Sato-Tate  :  $E_6$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 169,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.09049039083$
$L(\frac12)$  $\approx$  $0.09049039083$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 13$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 13$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad13$C_2$ \( 1 + 5 T + p T^{2} \)
good2$V_4$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$V_4$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
7$V_4$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$V_4$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$V_4$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$V_4$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$V_4$ \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
41$V_4$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$V_4$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$V_4$ \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$V_4$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$V_4$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$V_4$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−19.338464448, −18.7692599976, −18.5657697088, −17.6589598423, −17.2045770226, −16.6403222149, −16.5835711875, −14.9514079284, −14.9035789667, −13.0633955057, −12.5122422286, −11.4586174354, −10.8488398635, −9.91747700798, −9.56583207836, −8.4555260544, −7.54467569516, −6.57037775432, −5.06823463541, 5.06823463541, 6.57037775432, 7.54467569516, 8.4555260544, 9.56583207836, 9.91747700798, 10.8488398635, 11.4586174354, 12.5122422286, 13.0633955057, 14.9035789667, 14.9514079284, 16.5835711875, 16.6403222149, 17.2045770226, 17.6589598423, 18.5657697088, 18.7692599976, 19.338464448

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