Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \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 − 3·6-s − 7-s − 8-s + 6·9-s + 5·11-s + 3·12-s − 7·13-s + 14-s + 16-s + 3·17-s − 6·18-s − 8·19-s − 3·21-s − 5·22-s + 4·23-s − 3·24-s + 7·26-s + 9·27-s − 28-s + 2·29-s − 10·31-s − 32-s + 15·33-s − 3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 1.50·11-s + 0.866·12-s − 1.94·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.41·18-s − 1.83·19-s − 0.654·21-s − 1.06·22-s + 0.834·23-s − 0.612·24-s + 1.37·26-s + 1.73·27-s − 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s + 2.61·33-s − 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(102550\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 293\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{102550} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 102550,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.771679238$
$L(\frac12)$  $\approx$  $2.771679238$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
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

−13.84220952205276, −13.15083302376450, −12.87892715479296, −12.28555341546034, −11.89510727927939, −11.24140146432165, −10.48790813606241, −10.06986643909977, −9.573110520261329, −9.244892048284846, −8.921658763872429, −8.222313417247938, −7.964290641517378, −7.258730462209957, −6.771298900623158, −6.638673564086653, −5.602234477508861, −4.870050329430691, −4.120530606841316, −3.824726973110201, −2.964622775035487, −2.700445864519910, −1.867294996896895, −1.594799153382221, −0.4919466085491667, 0.4919466085491667, 1.594799153382221, 1.867294996896895, 2.700445864519910, 2.964622775035487, 3.824726973110201, 4.120530606841316, 4.870050329430691, 5.602234477508861, 6.638673564086653, 6.771298900623158, 7.258730462209957, 7.964290641517378, 8.222313417247938, 8.921658763872429, 9.244892048284846, 9.573110520261329, 10.06986643909977, 10.48790813606241, 11.24140146432165, 11.89510727927939, 12.28555341546034, 12.87892715479296, 13.15083302376450, 13.84220952205276

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