Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 29^{2} $
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  + 4-s + 2·7-s − 2·9-s − 8·13-s + 16-s − 10·25-s + 2·28-s − 6·29-s − 2·36-s + 3·49-s − 8·52-s + 12·53-s − 12·59-s − 4·63-s + 64-s − 8·67-s − 5·81-s − 12·83-s − 16·91-s − 10·100-s − 8·103-s + 24·107-s + 4·109-s + 2·112-s − 6·116-s + 16·117-s − 22·121-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.755·7-s − 2/3·9-s − 2.21·13-s + 1/4·16-s − 2·25-s + 0.377·28-s − 1.11·29-s − 1/3·36-s + 3/7·49-s − 1.10·52-s + 1.64·53-s − 1.56·59-s − 0.503·63-s + 1/8·64-s − 0.977·67-s − 5/9·81-s − 1.31·83-s − 1.67·91-s − 100-s − 0.788·103-s + 2.32·107-s + 0.383·109-s + 0.188·112-s − 0.557·116-s + 1.47·117-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164836 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(164836\)    =    \(2^{2} \cdot 7^{2} \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{164836} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 164836,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;29\}$,\[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( ( 1 - T )^{2} \)
29$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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\[\begin{aligned}L(s) = \prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.874952487363077123688486490575, −8.574673996165233070158070432652, −7.73899886874729111293892264969, −7.57571100088867902110310233811, −7.30118085295530868562669097184, −6.55315413557578547514079906546, −5.82169350283327601695140144520, −5.57928681742950427486583645839, −4.95666282602491657624262174992, −4.39532860206812970989814622572, −3.75399676073639417659089288658, −2.86651642125411249039840301372, −2.31158992149233556513632886331, −1.71520431274631175063040122688, 0, 1.71520431274631175063040122688, 2.31158992149233556513632886331, 2.86651642125411249039840301372, 3.75399676073639417659089288658, 4.39532860206812970989814622572, 4.95666282602491657624262174992, 5.57928681742950427486583645839, 5.82169350283327601695140144520, 6.55315413557578547514079906546, 7.30118085295530868562669097184, 7.57571100088867902110310233811, 7.73899886874729111293892264969, 8.574673996165233070158070432652, 8.874952487363077123688486490575

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