Properties

Degree 4
Conductor $ 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  − 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  :  induced by $\chi_{13} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 169,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.0904903$
$L(\frac12)$  $\approx$  $0.0904903$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−20.10703301806788866096607442711, −19.33846444803067811938292685952, −18.76925999761091455577670097918, −18.56576970884593164181156978722, −17.65895984226676047880460224981, −17.20457702255851851831821238873, −16.64032221486140008944897783369, −16.58357118750575786137645040830, −14.95140792841582142043623560609, −14.90357896669741344337333157041, −13.06339550571964159220878092128, −12.51224222861724419404676510427, −11.45861743544082710207892638661, −10.84883986345867375872722724058, −9.917477007979000496132305323081, −9.565832078364084782726879302558, −8.455526054396895167012626272507, −7.54467569516254146308765020154, −6.57037775431560330867930744613, −5.06823463540602925038855173519, 5.06823463540602925038855173519, 6.57037775431560330867930744613, 7.54467569516254146308765020154, 8.455526054396895167012626272507, 9.565832078364084782726879302558, 9.917477007979000496132305323081, 10.84883986345867375872722724058, 11.45861743544082710207892638661, 12.51224222861724419404676510427, 13.06339550571964159220878092128, 14.90357896669741344337333157041, 14.95140792841582142043623560609, 16.58357118750575786137645040830, 16.64032221486140008944897783369, 17.20457702255851851831821238873, 17.65895984226676047880460224981, 18.56576970884593164181156978722, 18.76925999761091455577670097918, 19.33846444803067811938292685952, 20.10703301806788866096607442711

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