Properties

Degree 4
Conductor $ 3^{4} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s + 8·8-s + 4·13-s − 7·16-s − 2·17-s − 8·19-s + 10·25-s − 8·26-s − 14·32-s + 4·34-s + 16·38-s − 8·43-s + 16·47-s − 2·49-s − 20·50-s − 4·52-s − 12·53-s + 24·59-s + 35·64-s + 24·67-s + 2·68-s + 8·76-s − 24·83-s + 16·86-s + 20·89-s − 32·94-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 2.82·8-s + 1.10·13-s − 7/4·16-s − 0.485·17-s − 1.83·19-s + 2·25-s − 1.56·26-s − 2.47·32-s + 0.685·34-s + 2.59·38-s − 1.21·43-s + 2.33·47-s − 2/7·49-s − 2.82·50-s − 0.554·52-s − 1.64·53-s + 3.12·59-s + 35/8·64-s + 2.93·67-s + 0.242·68-s + 0.917·76-s − 2.63·83-s + 1.72·86-s + 2.11·89-s − 3.30·94-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 23409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(23409\)    =    \(3^{4} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{23409} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 23409,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4140331733$
$L(\frac12)$  $\approx$  $0.4140331733$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3 \( 1 \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−10.67019276252676296635009348463, −10.02120320190170304188941308070, −9.700904361232117750200167263973, −8.803122490443163463083083304378, −8.614666728887835615841036671452, −8.525512840279562471351607821652, −7.73703964893348709714163440024, −6.94874426170867693671852723273, −6.55314321322967584637318361014, −5.54109626339049705766674438727, −4.82491262168358477569632493973, −4.26184654181768534650831433465, −3.60800496623101173067285949740, −2.10376560199305166100303049316, −0.864208652357659300161169448728, 0.864208652357659300161169448728, 2.10376560199305166100303049316, 3.60800496623101173067285949740, 4.26184654181768534650831433465, 4.82491262168358477569632493973, 5.54109626339049705766674438727, 6.55314321322967584637318361014, 6.94874426170867693671852723273, 7.73703964893348709714163440024, 8.525512840279562471351607821652, 8.614666728887835615841036671452, 8.803122490443163463083083304378, 9.700904361232117750200167263973, 10.02120320190170304188941308070, 10.67019276252676296635009348463

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