Properties

Degree 4
Conductor $ 13^{2} $
Sign $1$
Motivic weight 3
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 − 2·4-s − 6·6-s − 24·7-s − 15·8-s + 27·9-s + 24·11-s + 4·12-s + 91·13-s − 72·14-s − 15·16-s − 117·17-s + 81·18-s − 198·19-s + 48·21-s + 72·22-s + 78·23-s + 30·24-s + 247·25-s + 273·26-s − 154·27-s + 48·28-s + 141·29-s − 120·32-s − 48·33-s − 351·34-s + ⋯
L(s)  = 1  + 1.06·2-s − 0.384·3-s − 1/4·4-s − 0.408·6-s − 1.29·7-s − 0.662·8-s + 9-s + 0.657·11-s + 0.0962·12-s + 1.94·13-s − 1.37·14-s − 0.234·16-s − 1.66·17-s + 1.06·18-s − 2.39·19-s + 0.498·21-s + 0.697·22-s + 0.707·23-s + 0.255·24-s + 1.97·25-s + 2.05·26-s − 1.09·27-s + 0.323·28-s + 0.902·29-s − 0.662·32-s − 0.253·33-s − 1.77·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(169\)    =    \(13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{13} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 169,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.10692\)
\(L(\frac12)\)  \(\approx\)  \(1.10692\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 13$,\(F_p(T)\) is a polynomial of degree 4. If $p = 13$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ \( 1 - 7 p T + p^{3} T^{2} \)
good2$C_2^2$ \( 1 - 3 T + 11 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \)
3$C_2^2$ \( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 247 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 24 T + 535 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 24 T + 1523 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 117 T + 8776 T^{2} + 117 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 198 T + 19927 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 78 T - 6083 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 141 T - 4508 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 308 T + p^{3} T^{2} )( 1 + 308 T + p^{3} T^{2} ) \)
37$C_2^2$ \( 1 + 249 T + 71320 T^{2} + 249 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2^2$ \( 1 - 471 T + 142868 T^{2} - 471 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 104 T - 68691 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 116818 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 93 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 492 T + 286067 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 1362 T + 919111 T^{2} + 1362 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 1830 T + 1474211 T^{2} - 1830 p^{3} T^{3} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 567359 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 519766 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 1692 T + 1659257 T^{2} + 1692 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 348 T + 953041 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \)
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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

−19.50752909096018256887140678710, −19.38787102262126019114903646140, −18.38745997698692694960953777081, −18.00079251783951060487176370950, −16.92096272912926110469252774589, −16.39943340649917061283650070128, −15.52482972107435867423016421077, −15.14413292162704069430548515255, −13.97966085743690877963176414599, −13.36238127724265163184551261728, −12.76776063570069867432355301239, −12.62060054542997103347677841983, −10.94872441274391514249844106255, −10.67577508221439449057168113284, −9.145887368596140060017474345389, −8.751724509032825632034212394218, −6.55954385010205411763053889022, −6.44642247865807450236577898176, −4.61610486448213139359214098709, −3.82089278567738814603334571403, 3.82089278567738814603334571403, 4.61610486448213139359214098709, 6.44642247865807450236577898176, 6.55954385010205411763053889022, 8.751724509032825632034212394218, 9.145887368596140060017474345389, 10.67577508221439449057168113284, 10.94872441274391514249844106255, 12.62060054542997103347677841983, 12.76776063570069867432355301239, 13.36238127724265163184551261728, 13.97966085743690877963176414599, 15.14413292162704069430548515255, 15.52482972107435867423016421077, 16.39943340649917061283650070128, 16.92096272912926110469252774589, 18.00079251783951060487176370950, 18.38745997698692694960953777081, 19.38787102262126019114903646140, 19.50752909096018256887140678710

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