Properties

Degree 4
Conductor $ 17^{3} $
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 + 8·8-s − 6·9-s − 4·13-s − 7·16-s + 17-s + 12·18-s − 8·19-s − 6·25-s + 8·26-s − 14·32-s − 2·34-s + 6·36-s + 16·38-s + 8·43-s + 2·49-s + 12·50-s + 4·52-s + 12·53-s − 24·59-s + 35·64-s + 8·67-s − 68-s − 48·72-s + 8·76-s + 27·81-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s − 2·9-s − 1.10·13-s − 7/4·16-s + 0.242·17-s + 2.82·18-s − 1.83·19-s − 6/5·25-s + 1.56·26-s − 2.47·32-s − 0.342·34-s + 36-s + 2.59·38-s + 1.21·43-s + 2/7·49-s + 1.69·50-s + 0.554·52-s + 1.64·53-s − 3.12·59-s + 35/8·64-s + 0.977·67-s − 0.121·68-s − 5.65·72-s + 0.917·76-s + 3·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4913 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(4913\)    =    \(17^{3}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4913} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 4913,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 17$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 17$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad17$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−11.93623114442209313196485845061, −11.02941934245556844339132034347, −10.71734099889110000433050476633, −10.04009807089741531902747917178, −9.478907532433233918366672610314, −8.796016188301955158560715963145, −8.695686711870280818326611584019, −7.81910395523808377988054564239, −7.59089249699874684434940494125, −6.30307949005461764625435345620, −5.50273032332247760535071006049, −4.74199315541377008016560012773, −3.87810234950044466939896282289, −2.34450910132149137086379940903, 0, 2.34450910132149137086379940903, 3.87810234950044466939896282289, 4.74199315541377008016560012773, 5.50273032332247760535071006049, 6.30307949005461764625435345620, 7.59089249699874684434940494125, 7.81910395523808377988054564239, 8.695686711870280818326611584019, 8.796016188301955158560715963145, 9.478907532433233918366672610314, 10.04009807089741531902747917178, 10.71734099889110000433050476633, 11.02941934245556844339132034347, 11.93623114442209313196485845061

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