Properties

Degree 4
Conductor $ 2^{6} \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 6·7-s + 2·9-s − 12·14-s − 4·16-s + 4·17-s + 4·18-s + 8·23-s − 6·25-s − 12·28-s − 10·31-s − 8·32-s + 8·34-s + 4·36-s + 8·41-s + 16·46-s + 10·47-s + 14·49-s − 12·50-s − 20·62-s − 12·63-s − 8·64-s + 8·68-s + 12·71-s + 5·73-s + 8·79-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 2.26·7-s + 2/3·9-s − 3.20·14-s − 16-s + 0.970·17-s + 0.942·18-s + 1.66·23-s − 6/5·25-s − 2.26·28-s − 1.79·31-s − 1.41·32-s + 1.37·34-s + 2/3·36-s + 1.24·41-s + 2.35·46-s + 1.45·47-s + 2·49-s − 1.69·50-s − 2.54·62-s − 1.51·63-s − 64-s + 0.970·68-s + 1.42·71-s + 0.585·73-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4672\)    =    \(2^{6} \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4672} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4672,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.342826702$
$L(\frac12)$  $\approx$  $1.342826702$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;73\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;73\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good3$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
29$V_4$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$V_4$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$V_4$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
53$V_4$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
59$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
67$V_4$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$V_4$ \( 1 + 146 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.63974512870399257039650739060, −12.16551706224909237874193464731, −11.28268555030511406812863015604, −10.69724195694489126020802959948, −9.898328950839830403199017345697, −9.375040266738990391293249866098, −9.088421462106022162612758583749, −7.72922427819488925640567546189, −7.02428954566466171273305250222, −6.56249730409478760856133043539, −5.79621689921985941311583977855, −5.28234939584660765761721801828, −4.02676825708374336048445873000, −3.55176454243207444733428451920, −2.72454121493799864819769185152, 2.72454121493799864819769185152, 3.55176454243207444733428451920, 4.02676825708374336048445873000, 5.28234939584660765761721801828, 5.79621689921985941311583977855, 6.56249730409478760856133043539, 7.02428954566466171273305250222, 7.72922427819488925640567546189, 9.088421462106022162612758583749, 9.375040266738990391293249866098, 9.898328950839830403199017345697, 10.69724195694489126020802959948, 11.28268555030511406812863015604, 12.16551706224909237874193464731, 12.63974512870399257039650739060

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