Properties

Degree 4
Conductor $ 2^{6} \cdot 89 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 7-s + 4·9-s + 4·16-s + 6·17-s − 9·23-s − 7·25-s − 2·28-s − 2·31-s − 8·36-s − 3·41-s − 3·47-s − 11·49-s + 4·63-s − 8·64-s − 12·68-s + 15·71-s + 10·73-s − 2·79-s + 7·81-s − 10·89-s + 18·92-s + 16·97-s + 14·100-s − 17·103-s + 4·112-s − 9·113-s + ⋯
L(s)  = 1  − 4-s + 0.377·7-s + 4/3·9-s + 16-s + 1.45·17-s − 1.87·23-s − 7/5·25-s − 0.377·28-s − 0.359·31-s − 4/3·36-s − 0.468·41-s − 0.437·47-s − 1.57·49-s + 0.503·63-s − 64-s − 1.45·68-s + 1.78·71-s + 1.17·73-s − 0.225·79-s + 7/9·81-s − 1.05·89-s + 1.87·92-s + 1.62·97-s + 7/5·100-s − 1.67·103-s + 0.377·112-s − 0.846·113-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5696\)    =    \(2^{6} \cdot 89\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 5696,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.7901956560$
$L(\frac12)$  $\approx$  $0.7901956560$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;89\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;89\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
good3$V_4$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$V_4$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$V_4$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$V_4$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$V_4$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$V_4$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$V_4$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.20703924611948341688496641115, −11.71550216284689883563122841695, −10.84750641569332611743742059218, −10.09768613149996243646741210754, −9.770039584932557181905556987666, −9.480062514479180972974917534955, −8.317149012286868022316948731457, −7.984921525816833166850583301921, −7.48178097370659919750042110549, −6.47140897935593922836418326854, −5.66398479900295642288052860520, −4.98267490775586741991304360018, −4.10979129185551868010481677285, −3.58485015570827183215857421583, −1.70024136320843567528448310119, 1.70024136320843567528448310119, 3.58485015570827183215857421583, 4.10979129185551868010481677285, 4.98267490775586741991304360018, 5.66398479900295642288052860520, 6.47140897935593922836418326854, 7.48178097370659919750042110549, 7.984921525816833166850583301921, 8.317149012286868022316948731457, 9.480062514479180972974917534955, 9.770039584932557181905556987666, 10.09768613149996243646741210754, 10.84750641569332611743742059218, 11.71550216284689883563122841695, 12.20703924611948341688496641115

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