Properties

Degree 4
Conductor $ 2^{7} \cdot 3 \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s + 8·11-s − 12-s + 16-s − 8·17-s − 2·18-s − 19-s + 8·22-s − 24-s − 2·25-s + 2·27-s + 32-s − 8·33-s − 8·34-s − 2·36-s − 38-s − 4·41-s − 4·43-s + 8·44-s − 48-s + 2·49-s − 2·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 2.41·11-s − 0.288·12-s + 1/4·16-s − 1.94·17-s − 0.471·18-s − 0.229·19-s + 1.70·22-s − 0.204·24-s − 2/5·25-s + 0.384·27-s + 0.176·32-s − 1.39·33-s − 1.37·34-s − 1/3·36-s − 0.162·38-s − 0.624·41-s − 0.609·43-s + 1.20·44-s − 0.144·48-s + 2/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7296\)    =    \(2^{7} \cdot 3 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7296} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 7296,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.156426687$
$L(\frac12)$  $\approx$  $1.156426687$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good5$V_4$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$V_4$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$V_4$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
37$V_4$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$V_4$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$V_4$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
79$V_4$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \)
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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

−11.78621533917429449584146772905, −11.52980380136656255678675508354, −10.97678819419585092558064940921, −10.33021314014943111899681846904, −9.498534086647293987113947819835, −8.805995225296419929757103530190, −8.635109909037930511349667473118, −7.38500173081223226659754385270, −6.76551225484098574247524444367, −6.25428895436260465465645273423, −5.83156559761904980357839639162, −4.68758219507073195380503092750, −4.21769775255155791722086031431, −3.30499258423259072240785008617, −1.89359541125181195449265075444, 1.89359541125181195449265075444, 3.30499258423259072240785008617, 4.21769775255155791722086031431, 4.68758219507073195380503092750, 5.83156559761904980357839639162, 6.25428895436260465465645273423, 6.76551225484098574247524444367, 7.38500173081223226659754385270, 8.635109909037930511349667473118, 8.805995225296419929757103530190, 9.498534086647293987113947819835, 10.33021314014943111899681846904, 10.97678819419585092558064940921, 11.52980380136656255678675508354, 11.78621533917429449584146772905

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