Properties

Degree 4
Conductor $ 5^{3} \cdot 11 $
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 − 5-s − 2·9-s − 11-s − 3·16-s + 4·19-s + 20-s + 25-s + 4·31-s + 2·36-s + 44-s + 2·45-s + 14·49-s + 55-s − 12·59-s − 8·61-s + 7·64-s − 4·76-s + 4·79-s + 3·80-s − 5·81-s − 12·89-s − 4·95-s + 2·99-s − 100-s − 12·101-s + 4·109-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 3/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s + 0.718·31-s + 1/3·36-s + 0.150·44-s + 0.298·45-s + 2·49-s + 0.134·55-s − 1.56·59-s − 1.02·61-s + 7/8·64-s − 0.458·76-s + 0.450·79-s + 0.335·80-s − 5/9·81-s − 1.27·89-s − 0.410·95-s + 0.201·99-s − 0.0999·100-s − 1.19·101-s + 0.383·109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1375\)    =    \(5^{3} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1375} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 1375,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.4936014653$
$L(\frac12)$  $\approx$  $0.4936014653$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$ \( 1 + T \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
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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

−13.75539545875999856307279939656, −13.40790056481005511778602564463, −12.42556471409184263431561392223, −11.99318278851470879678465377104, −11.27748233626068839689230673523, −10.71705090685439114964224517084, −9.850315198815836374223293791035, −9.142886062799315513511972371368, −8.558267557755620760586313944365, −7.78942950669997960453513423290, −7.04063543577616078388107278171, −5.99080016242421450165909429669, −5.09698455816962605625731143596, −4.17584295292025542328208055314, −2.89395902106255541986149724624, 2.89395902106255541986149724624, 4.17584295292025542328208055314, 5.09698455816962605625731143596, 5.99080016242421450165909429669, 7.04063543577616078388107278171, 7.78942950669997960453513423290, 8.558267557755620760586313944365, 9.142886062799315513511972371368, 9.850315198815836374223293791035, 10.71705090685439114964224517084, 11.27748233626068839689230673523, 11.99318278851470879678465377104, 12.42556471409184263431561392223, 13.40790056481005511778602564463, 13.75539545875999856307279939656

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