Properties

Degree 4
Conductor $ 7^{3} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 7-s − 8·8-s − 2·9-s + 2·11-s − 2·14-s − 7·16-s − 4·18-s + 4·22-s − 8·23-s − 6·25-s + 28-s − 12·29-s + 14·32-s + 2·36-s − 12·37-s + 24·43-s − 2·44-s − 16·46-s + 49-s − 12·50-s − 12·53-s + 8·56-s − 24·58-s + 2·63-s + 35·64-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 0.377·7-s − 2.82·8-s − 2/3·9-s + 0.603·11-s − 0.534·14-s − 7/4·16-s − 0.942·18-s + 0.852·22-s − 1.66·23-s − 6/5·25-s + 0.188·28-s − 2.22·29-s + 2.47·32-s + 1/3·36-s − 1.97·37-s + 3.65·43-s − 0.301·44-s − 2.35·46-s + 1/7·49-s − 1.69·50-s − 1.64·53-s + 1.06·56-s − 3.15·58-s + 0.251·63-s + 35/8·64-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  :  no
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((4,\ 41503,\ (\ :1/2, 1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−9.757325112612361159806534316063, −9.519898606480484632296989725464, −8.920762498226765508675400216409, −8.678915900547504955372461266927, −7.82867506007281568614416512666, −7.37145602881524282201485228293, −6.25417326558836201502812232893, −5.96443978110022033983848919969, −5.62343675689709412781095073358, −4.94543315829774789302801165548, −4.07930883034575373640745113861, −3.86661484554997482031837898039, −3.26888344602981049454952325554, −2.20053354537635753960919690772, 0, 2.20053354537635753960919690772, 3.26888344602981049454952325554, 3.86661484554997482031837898039, 4.07930883034575373640745113861, 4.94543315829774789302801165548, 5.62343675689709412781095073358, 5.96443978110022033983848919969, 6.25417326558836201502812232893, 7.37145602881524282201485228293, 7.82867506007281568614416512666, 8.678915900547504955372461266927, 8.920762498226765508675400216409, 9.519898606480484632296989725464, 9.757325112612361159806534316063

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