Properties

Degree 4
Conductor $ 3 \cdot 11^{2} \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 3·5-s − 2·9-s + 2·11-s − 2·12-s − 3·15-s + 6·20-s − 12·23-s − 25-s − 2·27-s + 31-s + 2·33-s + 4·36-s − 14·37-s − 4·44-s + 6·45-s + 10·47-s − 5·49-s + 7·53-s − 6·55-s + 6·60-s + 8·64-s + 3·67-s − 12·69-s + 71-s − 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s + 0.603·11-s − 0.577·12-s − 0.774·15-s + 1.34·20-s − 2.50·23-s − 1/5·25-s − 0.384·27-s + 0.179·31-s + 0.348·33-s + 2/3·36-s − 2.30·37-s − 0.603·44-s + 0.894·45-s + 1.45·47-s − 5/7·49-s + 0.961·53-s − 0.809·55-s + 0.774·60-s + 64-s + 0.366·67-s − 1.44·69-s + 0.118·71-s − 0.115·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21417 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21417 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21417\)    =    \(3 \cdot 11^{2} \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{21417} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 21417,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;59\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11,\;59\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
good2$V_4$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$V_4$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$V_4$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
19$V_4$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$V_4$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$V_4$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$V_4$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$V_4$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$V_4$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
79$V_4$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−10.53689725368450372854599592141, −9.826822870483360437079190157580, −9.445421472012554194147888312809, −8.639862454686285533634005985397, −8.505212176467540948543859627828, −8.001364511684348274511215868827, −7.42540434027782802193190640285, −6.71321116389340966816178045003, −5.89217844221131912664123822930, −5.25273770817460594855381165623, −4.26184914397204860940102278366, −3.96514181338040128750591760867, −3.38791302940571504979288791714, −2.14227175602667477107494673109, 0, 2.14227175602667477107494673109, 3.38791302940571504979288791714, 3.96514181338040128750591760867, 4.26184914397204860940102278366, 5.25273770817460594855381165623, 5.89217844221131912664123822930, 6.71321116389340966816178045003, 7.42540434027782802193190640285, 8.001364511684348274511215868827, 8.505212176467540948543859627828, 8.639862454686285533634005985397, 9.445421472012554194147888312809, 9.826822870483360437079190157580, 10.53689725368450372854599592141

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