Properties

Degree 2
Conductor $ 29 \cdot 277 $
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 + 3-s − 4-s − 3·5-s + 6-s + 2·7-s − 3·8-s − 2·9-s − 3·10-s − 3·11-s − 12-s + 13-s + 2·14-s − 3·15-s − 16-s − 2·17-s − 2·18-s − 4·19-s + 3·20-s + 2·21-s − 3·22-s − 4·23-s − 3·24-s + 4·25-s + 26-s − 5·27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.755·7-s − 1.06·8-s − 2/3·9-s − 0.948·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.774·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.670·20-s + 0.436·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s + 4/5·25-s + 0.196·26-s − 0.962·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8033\)    =    \(29 \cdot 277\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8033} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8033,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9838230950$
$L(\frac12)$  $\approx$  $0.9838230950$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{29,\;277\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{29,\;277\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 \( 1 - T \)
277 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.029742706159624117640755861966, −7.37703461668037548252292425993, −6.29534025377701360004449313040, −5.55633080916977182348973231632, −4.89117299024782644097051405730, −4.10671859923880675297647366568, −3.77871505711256024954745440712, −2.87215146280125656419142596664, −2.10147262351322083877234448628, −0.40418153111815300411771914768, 0.40418153111815300411771914768, 2.10147262351322083877234448628, 2.87215146280125656419142596664, 3.77871505711256024954745440712, 4.10671859923880675297647366568, 4.89117299024782644097051405730, 5.55633080916977182348973231632, 6.29534025377701360004449313040, 7.37703461668037548252292425993, 8.029742706159624117640755861966

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