Properties

Degree 2
Conductor $ 3^{2} \cdot 5077 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 4·5-s − 4·7-s + 8·10-s + 6·11-s − 4·13-s − 8·14-s − 4·16-s + 4·17-s − 7·19-s + 8·20-s + 12·22-s + 6·23-s + 11·25-s − 8·26-s − 8·28-s + 6·29-s − 2·31-s − 8·32-s + 8·34-s − 16·35-s − 14·38-s − 8·43-s + 12·44-s + 12·46-s + 9·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s − 1.51·7-s + 2.52·10-s + 1.80·11-s − 1.10·13-s − 2.13·14-s − 16-s + 0.970·17-s − 1.60·19-s + 1.78·20-s + 2.55·22-s + 1.25·23-s + 11/5·25-s − 1.56·26-s − 1.51·28-s + 1.11·29-s − 0.359·31-s − 1.41·32-s + 1.37·34-s − 2.70·35-s − 2.27·38-s − 1.21·43-s + 1.80·44-s + 1.76·46-s + 1.31·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 45693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 45693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(45693\)    =    \(3^{2} \cdot 5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{45693} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 45693,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.838359529$
$L(\frac12)$  $\approx$  $6.838359529$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5077\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5077\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5077 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.56126870755163, −14.05021953982237, −13.56693925040079, −13.13829038993687, −12.73489120161547, −12.17879003079911, −12.01103688269687, −11.05885342406502, −10.27744978639475, −10.02539817947612, −9.408108433754635, −9.010798890653609, −8.614233567333854, −7.137554480062958, −6.772480740250183, −6.507090595799147, −5.880905547949235, −5.555077531192093, −4.798388036170816, −4.225828609152225, −3.523633469085458, −2.913656662784745, −2.442106717102284, −1.691708795020791, −0.7194774100373959, 0.7194774100373959, 1.691708795020791, 2.442106717102284, 2.913656662784745, 3.523633469085458, 4.225828609152225, 4.798388036170816, 5.555077531192093, 5.880905547949235, 6.507090595799147, 6.772480740250183, 7.137554480062958, 8.614233567333854, 9.010798890653609, 9.408108433754635, 10.02539817947612, 10.27744978639475, 11.05885342406502, 12.01103688269687, 12.17879003079911, 12.73489120161547, 13.13829038993687, 13.56693925040079, 14.05021953982237, 14.56126870755163

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