Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 7 \cdot 293 $
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 − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 5·11-s − 3·12-s + 7·13-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 8·19-s + 20-s − 3·21-s + 5·22-s − 4·23-s − 3·24-s + 25-s + 7·26-s − 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 1.50·11-s − 0.866·12-s + 1.94·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 1.83·19-s + 0.223·20-s − 0.654·21-s + 1.06·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.37·26-s − 1.73·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20510\)    =    \(2 \cdot 5 \cdot 7 \cdot 293\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20510} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 20510,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.410202351$
$L(\frac12)$  $\approx$  $2.410202351$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7,\;293\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7,\;293\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
293 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.80790532901775, −15.20605604877531, −14.36874426363443, −14.04285699208224, −13.32122658110430, −12.78507673521286, −12.33801515452189, −11.78991641648164, −11.15786745509098, −10.86221479323303, −10.55952892583002, −9.669109510219678, −8.756523676244383, −8.548714138043718, −7.304376585210975, −6.650825626673051, −6.338403572158896, −5.944400520363097, −5.325973148425190, −4.567260181747752, −3.963733817593297, −3.631467474122560, −1.906521739717156, −1.692459366668188, −0.6459308118531405, 0.6459308118531405, 1.692459366668188, 1.906521739717156, 3.631467474122560, 3.963733817593297, 4.567260181747752, 5.325973148425190, 5.944400520363097, 6.338403572158896, 6.650825626673051, 7.304376585210975, 8.548714138043718, 8.756523676244383, 9.669109510219678, 10.55952892583002, 10.86221479323303, 11.15786745509098, 11.78991641648164, 12.33801515452189, 12.78507673521286, 13.32122658110430, 14.04285699208224, 14.36874426363443, 15.20605604877531, 15.80790532901775

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