Properties

Degree 4
Conductor $ 3^{2} \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 − 9-s + 4·13-s − 7·16-s + 2·17-s − 2·18-s − 8·19-s + 10·25-s + 8·26-s + 14·32-s + 4·34-s + 36-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 + 8·72-s + 8·76-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s − 1/3·9-s + 1.10·13-s − 7/4·16-s + 0.485·17-s − 0.471·18-s − 1.83·19-s + 2·25-s + 1.56·26-s + 2.47·32-s + 0.685·34-s + 1/6·36-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 + 0.942·72-s + 0.917·76-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2601\)    =    \(3^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2601} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2601,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9694707910$
$L(\frac12)$  $\approx$  $0.9694707910$
$L(\frac{3}{2})$   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$C_2$ \( 1 + T^{2} \)
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

−13.22313622696751046955945037970, −12.48071732359980117615625341463, −12.29589703833658958394204491211, −11.29197683349063617608596857547, −10.77052143340783570038150502125, −9.868263186905898811129796321835, −9.170445946119745654646751668784, −8.397074917737072518802833061281, −8.353938731587443077985698253799, −6.58665318563334711739225264874, −6.25244206026251981780374239300, −5.25757798355882877965671943400, −4.72410306496764483411551904974, −3.83132277647941273193838504179, −3.08123160264227815861676586344, 3.08123160264227815861676586344, 3.83132277647941273193838504179, 4.72410306496764483411551904974, 5.25757798355882877965671943400, 6.25244206026251981780374239300, 6.58665318563334711739225264874, 8.353938731587443077985698253799, 8.397074917737072518802833061281, 9.170445946119745654646751668784, 9.868263186905898811129796321835, 10.77052143340783570038150502125, 11.29197683349063617608596857547, 12.29589703833658958394204491211, 12.48071732359980117615625341463, 13.22313622696751046955945037970

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