Properties

Degree 4
Conductor $ 3^{2} \cdot 379 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s − 6·7-s + 9-s + 6·12-s + 5·13-s + 5·16-s − 9·19-s + 12·21-s + 25-s + 4·27-s + 18·28-s + 2·31-s − 3·36-s − 16·37-s − 10·39-s − 3·43-s − 10·48-s + 17·49-s − 15·52-s + 18·57-s − 6·61-s − 6·63-s − 3·64-s − 67-s + 5·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 2.26·7-s + 1/3·9-s + 1.73·12-s + 1.38·13-s + 5/4·16-s − 2.06·19-s + 2.61·21-s + 1/5·25-s + 0.769·27-s + 3.40·28-s + 0.359·31-s − 1/2·36-s − 2.63·37-s − 1.60·39-s − 0.457·43-s − 1.44·48-s + 17/7·49-s − 2.08·52-s + 2.38·57-s − 0.768·61-s − 0.755·63-s − 3/8·64-s − 0.122·67-s + 0.585·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3411 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3411 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3411\)    =    \(3^{2} \cdot 379\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{3411} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 3411,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;379\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;379\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + 2 T + p T^{2} \)
379$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 11 T + p T^{2} ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \)
17$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.52904316215616694165779948272, −12.05468628216561216405853219287, −11.02103899512350083464916962731, −10.47881770793489996875163931921, −10.09884140552019139191044688156, −9.298818815832352824793294835629, −8.761020043918534300736028943085, −8.368184913151009783371565215147, −6.84996672400503775780963825268, −6.41170868431526283729680636959, −5.89846952311957618693090452483, −5.01094266926562858529869899609, −4.03936375515308084295939588309, −3.34254052952856063975225663394, 0, 3.34254052952856063975225663394, 4.03936375515308084295939588309, 5.01094266926562858529869899609, 5.89846952311957618693090452483, 6.41170868431526283729680636959, 6.84996672400503775780963825268, 8.368184913151009783371565215147, 8.761020043918534300736028943085, 9.298818815832352824793294835629, 10.09884140552019139191044688156, 10.47881770793489996875163931921, 11.02103899512350083464916962731, 12.05468628216561216405853219287, 12.52904316215616694165779948272

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