Properties

Degree 4
Conductor $ 2^{7} \cdot 17^{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  + 2-s + 4-s − 8·7-s + 8-s − 2·9-s − 8·14-s + 16-s − 2·17-s − 2·18-s − 10·25-s − 8·28-s − 8·31-s + 32-s − 2·34-s − 2·36-s + 12·41-s + 34·49-s − 10·50-s − 8·56-s − 8·62-s + 16·63-s + 64-s − 2·68-s − 2·72-s + 4·73-s + 16·79-s − 5·81-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 3.02·7-s + 0.353·8-s − 2/3·9-s − 2.13·14-s + 1/4·16-s − 0.485·17-s − 0.471·18-s − 2·25-s − 1.51·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/3·36-s + 1.87·41-s + 34/7·49-s − 1.41·50-s − 1.06·56-s − 1.01·62-s + 2.01·63-s + 1/8·64-s − 0.242·68-s − 0.235·72-s + 0.468·73-s + 1.80·79-s − 5/9·81-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(36992\)    =    \(2^{7} \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{36992} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 36992,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;17\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 - T \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−9.866374203841653278096661486834, −9.628355001583189625988551593246, −9.286071988264369821451329617999, −8.657255224248218342106471033370, −7.76589645819579710746731573989, −7.25946473788827804639135121523, −6.65330731261113455462227374582, −6.03835056135927571761510178023, −5.99911855171913780844071238816, −5.19013856842488788481068619038, −3.90229547122613769308947697962, −3.78997975332835388697662085523, −2.95627382514208231650511213543, −2.34610746208050944602353534104, 0, 2.34610746208050944602353534104, 2.95627382514208231650511213543, 3.78997975332835388697662085523, 3.90229547122613769308947697962, 5.19013856842488788481068619038, 5.99911855171913780844071238816, 6.03835056135927571761510178023, 6.65330731261113455462227374582, 7.25946473788827804639135121523, 7.76589645819579710746731573989, 8.657255224248218342106471033370, 9.286071988264369821451329617999, 9.628355001583189625988551593246, 9.866374203841653278096661486834

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