Properties

Degree 4
Conductor $ 7^{3} \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 7-s − 2·9-s + 2·13-s − 3·16-s + 6·17-s + 8·19-s + 2·25-s + 28-s + 2·36-s − 8·37-s + 6·41-s + 49-s − 2·52-s + 12·53-s + 2·61-s + 2·63-s + 7·64-s + 16·67-s − 6·68-s − 10·73-s − 8·76-s − 5·81-s + 12·83-s − 2·91-s − 2·100-s + 18·101-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.377·7-s − 2/3·9-s + 0.554·13-s − 3/4·16-s + 1.45·17-s + 1.83·19-s + 2/5·25-s + 0.188·28-s + 1/3·36-s − 1.31·37-s + 0.937·41-s + 1/7·49-s − 0.277·52-s + 1.64·53-s + 0.256·61-s + 0.251·63-s + 7/8·64-s + 1.95·67-s − 0.727·68-s − 1.17·73-s − 0.917·76-s − 5/9·81-s + 1.31·83-s − 0.209·91-s − 1/5·100-s + 1.79·101-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 41503 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(41503\)    =    \(7^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{41503} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 41503,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.114337095$
$L(\frac12)$  $\approx$  $1.114337095$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ \( 1 + T \)
11$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−10.08413165334841815330474659559, −9.749659497981391797203883443975, −9.234516000430137164523456704711, −8.667044376361886927783253520503, −8.365969422532131255143167817892, −7.46527880275731058373331702716, −7.26928094611484628076589803252, −6.43906336777089411657390297016, −5.75141901363088440822485038831, −5.35388325492838072186666256000, −4.77442524762674841784226821531, −3.71668907115242125030065328458, −3.38112485638183544262061635605, −2.47131910765623514659467569057, −1.00416546591001491474959424526, 1.00416546591001491474959424526, 2.47131910765623514659467569057, 3.38112485638183544262061635605, 3.71668907115242125030065328458, 4.77442524762674841784226821531, 5.35388325492838072186666256000, 5.75141901363088440822485038831, 6.43906336777089411657390297016, 7.26928094611484628076589803252, 7.46527880275731058373331702716, 8.365969422532131255143167817892, 8.667044376361886927783253520503, 9.234516000430137164523456704711, 9.749659497981391797203883443975, 10.08413165334841815330474659559

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