Properties

Degree 4
Conductor $ 7 \cdot 29^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s + 7-s − 4·9-s − 2·13-s − 3·16-s − 4·20-s + 2·23-s + 2·25-s − 28-s + 4·29-s + 4·35-s + 4·36-s − 16·45-s − 4·49-s + 2·52-s + 12·53-s + 2·59-s − 4·63-s + 7·64-s − 8·65-s − 14·67-s − 12·80-s + 7·81-s − 12·83-s − 2·91-s − 2·92-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s + 0.377·7-s − 4/3·9-s − 0.554·13-s − 3/4·16-s − 0.894·20-s + 0.417·23-s + 2/5·25-s − 0.188·28-s + 0.742·29-s + 0.676·35-s + 2/3·36-s − 2.38·45-s − 4/7·49-s + 0.277·52-s + 1.64·53-s + 0.260·59-s − 0.503·63-s + 7/8·64-s − 0.992·65-s − 1.71·67-s − 1.34·80-s + 7/9·81-s − 1.31·83-s − 0.209·91-s − 0.208·92-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5887\)    =    \(7 \cdot 29^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5887} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 5887,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.9419312153$
$L(\frac12)$  $\approx$  $0.9419312153$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;29\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{7,\;29\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
29$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
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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

−11.83339669180257609565944302858, −11.68535507013654368859848359878, −10.81692738288498619970753615675, −10.23436050746063358406234314273, −9.712870704337976178143515135088, −9.126714636661016225410267328789, −8.733856836143840569093231414060, −8.043549391330799288857236799773, −7.10157501554749244505029850065, −6.30408672236389073866960478832, −5.68781909503681754640411478581, −5.22140645748655013566237515271, −4.35932945137902958767924049257, −2.90866313208207034385166985163, −2.05561271035874881637416218836, 2.05561271035874881637416218836, 2.90866313208207034385166985163, 4.35932945137902958767924049257, 5.22140645748655013566237515271, 5.68781909503681754640411478581, 6.30408672236389073866960478832, 7.10157501554749244505029850065, 8.043549391330799288857236799773, 8.733856836143840569093231414060, 9.126714636661016225410267328789, 9.712870704337976178143515135088, 10.23436050746063358406234314273, 10.81692738288498619970753615675, 11.68535507013654368859848359878, 11.83339669180257609565944302858

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