Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7 \cdot 293 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

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 − 3·12-s + 2·13-s + 14-s + 16-s − 7·17-s − 6·18-s + 4·19-s + 3·21-s + 4·23-s + 3·24-s − 2·26-s − 9·27-s − 28-s + 2·29-s + 31-s − 32-s + 7·34-s + 6·36-s + 2·37-s − 4·38-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 − 0.866·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 1.41·18-s + 0.917·19-s + 0.654·21-s + 0.834·23-s + 0.612·24-s − 0.392·26-s − 1.73·27-s − 0.188·28-s + 0.371·29-s + 0.179·31-s − 0.176·32-s + 1.20·34-s + 36-s + 0.328·37-s − 0.648·38-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  =  1
Selberg data  =  $(2,\ 102550,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−13.78110507055972, −13.25957500379916, −12.94423913834934, −12.36023187186725, −11.82794308560359, −11.33799137248323, −11.16760432685833, −10.59403862473490, −10.18296812719879, −9.601294872010812, −9.148891703789073, −8.589415688838006, −7.957394366432259, −7.255657152018911, −6.740362905134802, −6.565607803805177, −5.921537798220763, −5.452539085296752, −4.781703203680763, −4.403549465639312, −3.566950861722589, −2.884671949216451, −2.033739582551395, −1.295410016636226, −0.6988533233661706, 0, 0.6988533233661706, 1.295410016636226, 2.033739582551395, 2.884671949216451, 3.566950861722589, 4.403549465639312, 4.781703203680763, 5.452539085296752, 5.921537798220763, 6.565607803805177, 6.740362905134802, 7.255657152018911, 7.957394366432259, 8.589415688838006, 9.148891703789073, 9.601294872010812, 10.18296812719879, 10.59403862473490, 11.16760432685833, 11.33799137248323, 11.82794308560359, 12.36023187186725, 12.94423913834934, 13.25957500379916, 13.78110507055972

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