Properties

Label 2-50-1.1-c1-0-1
Degree $2$
Conductor $50$
Sign $1$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
Motivic weight $1$
Arithmetic yes
Rational yes
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

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9568112265\)
\(L(\frac12)\) \(\approx\) \(0.9568112265\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.85587358322534319659703271336, −14.27543395311716649730892535101, −13.31818220322005518851751496924, −12.18462032318833367073561378768, −11.16007422368021302024475397710, −9.930723730317650982781533521356, −8.071729218562685355881418106440, −6.35306147256574072432179047412, −5.34945783064988058695168525193, −3.29699975126484092788987476482, 3.29699975126484092788987476482, 5.34945783064988058695168525193, 6.35306147256574072432179047412, 8.071729218562685355881418106440, 9.930723730317650982781533521356, 11.16007422368021302024475397710, 12.18462032318833367073561378768, 13.31818220322005518851751496924, 14.27543395311716649730892535101, 15.85587358322534319659703271336

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