Properties

Degree 2
Conductor $ 2 \cdot 5^{2} $
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-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s − 3·11-s + 12-s − 4·13-s − 2·14-s + 16-s − 3·17-s + 2·18-s + 5·19-s + 2·21-s + 3·22-s + 6·23-s − 24-s + 4·26-s − 5·27-s + 2·28-s + 2·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 1.14·19-s + 0.436·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.962·27-s + 0.377·28-s + 0.359·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(50\)    =    \(2 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{50} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 50,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7131649814$
$L(\frac12)$  $\approx$  $0.7131649814$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.85985804066716, −18.75556473530924, −17.73623770272490, −16.90369970531125, −15.49987605150771, −14.64380916664077, −13.46639701582307, −11.89949732310571, −10.81431448769284, −9.447991916742082, −8.316812877075225, −7.298531347937641, −5.205921487885761, −2.651550404172352, 2.651550404172352, 5.205921487885761, 7.298531347937641, 8.316812877075225, 9.447991916742082, 10.81431448769284, 11.89949732310571, 13.46639701582307, 14.64380916664077, 15.49987605150771, 16.90369970531125, 17.73623770272490, 18.75556473530924, 19.85985804066716

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