Properties

Degree 2
Conductor $ 7^{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 + 3·3-s + 2·4-s + 4·5-s − 6·6-s + 6·9-s − 8·10-s − 6·11-s + 6·12-s + 4·13-s + 12·15-s − 4·16-s + 4·17-s − 12·18-s + 7·19-s + 8·20-s + 12·22-s − 6·23-s + 11·25-s − 8·26-s + 9·27-s − 6·29-s − 24·30-s + 2·31-s + 8·32-s − 18·33-s − 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 4-s + 1.78·5-s − 2.44·6-s + 2·9-s − 2.52·10-s − 1.80·11-s + 1.73·12-s + 1.10·13-s + 3.09·15-s − 16-s + 0.970·17-s − 2.82·18-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 + 1.73·27-s − 1.11·29-s − 4.38·30-s + 0.359·31-s + 1.41·32-s − 3.13·33-s − 1.37·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 248773 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 248773 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(248773\)    =    \(7^{2} \cdot 5077\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{248773} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 248773,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.476757167$
$L(\frac12)$  $\approx$  $4.476757167$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;5077\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;5077\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad7 \( 1 \)
5077 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
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

−13.21495713217107, −12.60174210332792, −11.91651749001402, −11.11899785196956, −10.64597468049235, −10.13797064707568, −9.986775786905225, −9.496743052984339, −9.309489571212765, −8.626136754440598, −8.246639038746404, −7.964867690579693, −7.386016565777356, −7.124755930679463, −6.256111629003216, −5.734638586791581, −5.312800747384428, −4.710552736043915, −3.788673222306224, −3.252611562616229, −2.724731424248563, −2.274584189815714, −1.724627432408949, −1.387773402928849, −0.6364604858554898, 0.6364604858554898, 1.387773402928849, 1.724627432408949, 2.274584189815714, 2.724731424248563, 3.252611562616229, 3.788673222306224, 4.710552736043915, 5.312800747384428, 5.734638586791581, 6.256111629003216, 7.124755930679463, 7.386016565777356, 7.964867690579693, 8.246639038746404, 8.626136754440598, 9.309489571212765, 9.496743052984339, 9.986775786905225, 10.13797064707568, 10.64597468049235, 11.11899785196956, 11.91651749001402, 12.60174210332792, 13.21495713217107

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