Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 7-s − 2·9-s + 12-s − 7·13-s + 16-s + 5·19-s − 21-s + 25-s + 5·27-s − 28-s + 8·31-s + 2·36-s + 37-s + 7·39-s − 2·43-s − 48-s − 7·49-s + 7·52-s − 5·57-s − 61-s − 2·63-s − 64-s − 14·67-s − 7·73-s − 75-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 0.377·7-s − 2/3·9-s + 0.288·12-s − 1.94·13-s + 1/4·16-s + 1.14·19-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 0.188·28-s + 1.43·31-s + 1/3·36-s + 0.164·37-s + 1.12·39-s − 0.304·43-s − 0.144·48-s − 49-s + 0.970·52-s − 0.662·57-s − 0.128·61-s − 0.251·63-s − 1/8·64-s − 1.71·67-s − 0.819·73-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1116\)    =    \(2^{2} \cdot 3^{2} \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1116} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1116,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4254283817$
$L(\frac12)$  $\approx$  $0.4254283817$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;31\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;31\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 7 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
83$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−14.08680482312613066827079930668, −13.57861727831866455994748310876, −12.70215991683975821083442252589, −11.94883798073866020533120603588, −11.80628704938984376814637709077, −10.89809460716251272126837437428, −10.06423424076045358628708764026, −9.599559494960929771349910383163, −8.718072445424649794509158417192, −7.899472857564502415835684557922, −7.19704267596062297832151605893, −6.12887186529772388269728794943, −5.15202294391188716329856472167, −4.65762024378348674735850325026, −2.92009081916655404042712155697, 2.92009081916655404042712155697, 4.65762024378348674735850325026, 5.15202294391188716329856472167, 6.12887186529772388269728794943, 7.19704267596062297832151605893, 7.899472857564502415835684557922, 8.718072445424649794509158417192, 9.599559494960929771349910383163, 10.06423424076045358628708764026, 10.89809460716251272126837437428, 11.80628704938984376814637709077, 11.94883798073866020533120603588, 12.70215991683975821083442252589, 13.57861727831866455994748310876, 14.08680482312613066827079930668

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