Properties

Degree 4
Conductor $ 13^{2} \cdot 31^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s − 6·5-s − 2·7-s − 5·8-s + 3·9-s + 6·10-s − 5·13-s + 2·14-s + 5·16-s − 5·17-s − 3·18-s − 4·19-s − 12·20-s − 4·23-s + 17·25-s + 5·26-s − 4·28-s − 3·29-s − 2·31-s − 10·32-s + 5·34-s + 12·35-s + 6·36-s + 5·37-s + 4·38-s + 30·40-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 2.68·5-s − 0.755·7-s − 1.76·8-s + 9-s + 1.89·10-s − 1.38·13-s + 0.534·14-s + 5/4·16-s − 1.21·17-s − 0.707·18-s − 0.917·19-s − 2.68·20-s − 0.834·23-s + 17/5·25-s + 0.980·26-s − 0.755·28-s − 0.557·29-s − 0.359·31-s − 1.76·32-s + 0.857·34-s + 2.02·35-s + 36-s + 0.821·37-s + 0.648·38-s + 4.74·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(162409\)    =    \(13^{2} \cdot 31^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{403} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(2\)
Selberg data  =  \((4,\ 162409,\ (\ :1/2, 1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;31\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad13$C_2$ \( 1 + 5 T + p T^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 4 T - 55 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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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

−11.11568724695176633663997251758, −10.70945340154624265979136607798, −9.996200631951439651012765452789, −9.804730765849498770748622836713, −8.992638168339053293765512103435, −8.880855987920655196567674472333, −8.067710430677879380414335847624, −7.75861698098793931903459896864, −7.49431526448001500136698274360, −6.78037284061726580088351460140, −6.67494376083675504444709883359, −6.08134337922909349089170455968, −5.02757091133561836856995891626, −4.42270367648091092903441358103, −3.94437873180279871162996668980, −3.43795209240806247331391329874, −2.77490665217558160198118340840, −1.96205391458802541652082003209, 0, 0, 1.96205391458802541652082003209, 2.77490665217558160198118340840, 3.43795209240806247331391329874, 3.94437873180279871162996668980, 4.42270367648091092903441358103, 5.02757091133561836856995891626, 6.08134337922909349089170455968, 6.67494376083675504444709883359, 6.78037284061726580088351460140, 7.49431526448001500136698274360, 7.75861698098793931903459896864, 8.067710430677879380414335847624, 8.880855987920655196567674472333, 8.992638168339053293765512103435, 9.804730765849498770748622836713, 9.996200631951439651012765452789, 10.70945340154624265979136607798, 11.11568724695176633663997251758

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