Properties

Degree 4
Conductor $ 2^{5} \cdot 5 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 4·9-s + 10-s − 3·13-s + 16-s − 6·17-s − 4·18-s − 20-s − 4·25-s + 3·26-s − 6·29-s − 32-s + 6·34-s + 4·36-s + 4·37-s + 40-s − 4·45-s + 14·49-s + 4·50-s − 3·52-s + 6·58-s − 2·61-s + 64-s + 3·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 4/3·9-s + 0.316·10-s − 0.832·13-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 0.223·20-s − 4/5·25-s + 0.588·26-s − 1.11·29-s − 0.176·32-s + 1.02·34-s + 2/3·36-s + 0.657·37-s + 0.158·40-s − 0.596·45-s + 2·49-s + 0.565·50-s − 0.416·52-s + 0.787·58-s − 0.256·61-s + 1/8·64-s + 0.372·65-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2080\)    =    \(2^{5} \cdot 5 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2080} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 2080,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4803631948$
$L(\frac12)$  $\approx$  $0.4803631948$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−13.20859068784444078804919596956, −12.59086967665366017226981995880, −12.01019791255937568083245043080, −11.34656975509688312402616161032, −10.74781547782242050869487639672, −10.08922399836353334407924537776, −9.480918778573358190832723629444, −8.933390642891826377839652184385, −8.004172495143742733490166513119, −7.34239524010294972057117625494, −6.93300330028434354687859432943, −5.91298812586149253328721774965, −4.68473046101937618803253601661, −3.89295282825188061140108988893, −2.19543188028031805089419316236, 2.19543188028031805089419316236, 3.89295282825188061140108988893, 4.68473046101937618803253601661, 5.91298812586149253328721774965, 6.93300330028434354687859432943, 7.34239524010294972057117625494, 8.004172495143742733490166513119, 8.933390642891826377839652184385, 9.480918778573358190832723629444, 10.08922399836353334407924537776, 10.74781547782242050869487639672, 11.34656975509688312402616161032, 12.01019791255937568083245043080, 12.59086967665366017226981995880, 13.20859068784444078804919596956

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