Properties

Degree 4
Conductor $ 2 \cdot 7^{2} \cdot 43 $
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 − 4-s − 7-s + 8-s + 4·9-s + 6·11-s + 14-s + 3·16-s − 4·18-s − 6·22-s − 9·23-s − 25-s + 28-s − 3·32-s − 4·36-s − 14·37-s + 9·43-s − 6·44-s + 9·46-s − 6·49-s + 50-s − 12·53-s − 56-s − 4·63-s − 5·64-s + 10·67-s + 21·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 1.80·11-s + 0.267·14-s + 3/4·16-s − 0.942·18-s − 1.27·22-s − 1.87·23-s − 1/5·25-s + 0.188·28-s − 0.530·32-s − 2/3·36-s − 2.30·37-s + 1.37·43-s − 0.904·44-s + 1.32·46-s − 6/7·49-s + 0.141·50-s − 1.64·53-s − 0.133·56-s − 0.503·63-s − 5/8·64-s + 1.22·67-s + 2.49·71-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4214\)    =    \(2 \cdot 7^{2} \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4214} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 4214,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5412741631$
$L(\frac12)$  $\approx$  $0.5412741631$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;43\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;43\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
7$C_2$ \( 1 + T + p T^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 8 T + p T^{2} ) \)
good3$V_4$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$V_4$ \( 1 + 37 T^{2} + p^{2} T^{4} \)
47$V_4$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$V_4$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
61$V_4$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
73$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$V_4$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
89$V_4$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
97$V_4$ \( 1 - 41 T^{2} + p^{2} T^{4} \)
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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

−12.40992743119000219830426930703, −12.02493042028340985141242906119, −11.19494436916396067256088003266, −10.41052341605502377252150880437, −9.844829833303857857340526608476, −9.505931410169624923239388157236, −8.994621533683078515979595306289, −8.227068586557118849738869505629, −7.62445480258900598893734797038, −6.71059934653250726444651732388, −6.31644324564184210493440834356, −5.19765492894545924172017700826, −4.07724382311546908680728359391, −3.71331134171430874660508363364, −1.61452194687764831160312520057, 1.61452194687764831160312520057, 3.71331134171430874660508363364, 4.07724382311546908680728359391, 5.19765492894545924172017700826, 6.31644324564184210493440834356, 6.71059934653250726444651732388, 7.62445480258900598893734797038, 8.227068586557118849738869505629, 8.994621533683078515979595306289, 9.505931410169624923239388157236, 9.844829833303857857340526608476, 10.41052341605502377252150880437, 11.19494436916396067256088003266, 12.02493042028340985141242906119, 12.40992743119000219830426930703

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