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Properties

Label 2-500-20.19-c0-0-2
Degree $2$
Conductor $500$
Sign $1$
Analytic cond. $0.249532$
Root an. cond. $0.499532$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.61·3-s + 4-s − 1.61·6-s − 0.618·7-s − 8-s + 1.61·9-s + 1.61·12-s + 0.618·14-s + 16-s − 1.61·18-s − 1.00·21-s − 0.618·23-s − 1.61·24-s + 27-s − 0.618·28-s − 1.61·29-s − 32-s + 1.61·36-s − 1.61·41-s + 1.00·42-s + 1.61·43-s + 0.618·46-s + 1.61·47-s + 1.61·48-s − 0.618·49-s − 54-s + ⋯
L(s)  = 1  − 2-s + 1.61·3-s + 4-s − 1.61·6-s − 0.618·7-s − 8-s + 1.61·9-s + 1.61·12-s + 0.618·14-s + 16-s − 1.61·18-s − 1.00·21-s − 0.618·23-s − 1.61·24-s + 27-s − 0.618·28-s − 1.61·29-s − 32-s + 1.61·36-s − 1.61·41-s + 1.00·42-s + 1.61·43-s + 0.618·46-s + 1.61·47-s + 1.61·48-s − 0.618·49-s − 54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(500\)    =    \(2^{2} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(0.249532\)
Root analytic conductor: \(0.499532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{500} (499, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8974930687\)
\(L(\frac12)\) \(\approx\) \(0.8974930687\)
\(L(1)\) 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 \( 1 - 1.61T + T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 - 1.61T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.618T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.79009569541155256020998785666, −9.892053578804294027684938270086, −9.269216966860930361686480550770, −8.615762592868283300193694511914, −7.72457624057647123432192311638, −7.07167899836745740199562808627, −5.85287572927177250505680003377, −3.90338335502008625780572165396, −2.95455403281289193232370231576, −1.90018583643744816330930253607, 1.90018583643744816330930253607, 2.95455403281289193232370231576, 3.90338335502008625780572165396, 5.85287572927177250505680003377, 7.07167899836745740199562808627, 7.72457624057647123432192311638, 8.615762592868283300193694511914, 9.269216966860930361686480550770, 9.892053578804294027684938270086, 10.79009569541155256020998785666

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