Properties

Degree 4
Conductor $ 7^{2} \cdot 863^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s − 4-s + 5-s − 6·6-s − 2·7-s + 8·8-s + 2·9-s − 2·10-s − 3·11-s − 3·12-s + 9·13-s + 4·14-s + 3·15-s − 7·16-s − 2·17-s − 4·18-s + 3·19-s − 20-s − 6·21-s + 6·22-s − 23-s + 24·24-s − 8·25-s − 18·26-s − 6·27-s + 2·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s − 1/2·4-s + 0.447·5-s − 2.44·6-s − 0.755·7-s + 2.82·8-s + 2/3·9-s − 0.632·10-s − 0.904·11-s − 0.866·12-s + 2.49·13-s + 1.06·14-s + 0.774·15-s − 7/4·16-s − 0.485·17-s − 0.942·18-s + 0.688·19-s − 0.223·20-s − 1.30·21-s + 1.27·22-s − 0.208·23-s + 4.89·24-s − 8/5·25-s − 3.53·26-s − 1.15·27-s + 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36493681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36493681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36493681\)    =    \(7^{2} \cdot 863^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6041} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 36493681,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;863\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;863\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$ \( ( 1 + T )^{2} \)
863$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 3 T + 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 19 T + 207 T^{2} + 19 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 21 T + 233 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 3 T + 137 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 25 T + 313 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 3 T + 185 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
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

−7.966164147079045004430370908122, −7.903593883955323280920483257250, −7.42825530040303276910612644853, −7.24082249208048815642605076363, −6.54443210285323835346686876076, −6.19755158783371892609049136812, −5.61967847425132467550273266263, −5.60317516235288202814959805631, −5.04506575913172768637878753450, −4.48049639839874853547067916470, −3.92688796547235443293728004325, −3.80619497765751653309732269049, −3.29277313806043495190102038987, −3.21592732367449984372972918221, −2.29510876803931096661565979706, −2.12493957335690480931053399602, −1.37755701035714148138910374502, −1.22875976686850573077630225103, 0, 0, 1.22875976686850573077630225103, 1.37755701035714148138910374502, 2.12493957335690480931053399602, 2.29510876803931096661565979706, 3.21592732367449984372972918221, 3.29277313806043495190102038987, 3.80619497765751653309732269049, 3.92688796547235443293728004325, 4.48049639839874853547067916470, 5.04506575913172768637878753450, 5.60317516235288202814959805631, 5.61967847425132467550273266263, 6.19755158783371892609049136812, 6.54443210285323835346686876076, 7.24082249208048815642605076363, 7.42825530040303276910612644853, 7.903593883955323280920483257250, 7.966164147079045004430370908122

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