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Properties

Label 2-500-4.3-c0-0-0
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 1.61i·3-s − 4-s − 1.61·6-s + 0.618i·7-s − i·8-s − 1.61·9-s − 1.61i·12-s − 0.618·14-s + 16-s − 1.61i·18-s − 1.00·21-s − 0.618i·23-s + 1.61·24-s − i·27-s − 0.618i·28-s + ⋯
L(s)  = 1  + i·2-s + 1.61i·3-s − 4-s − 1.61·6-s + 0.618i·7-s − i·8-s − 1.61·9-s − 1.61i·12-s − 0.618·14-s + 16-s − 1.61i·18-s − 1.00·21-s − 0.618i·23-s + 1.61·24-s − i·27-s − 0.618i·28-s + ⋯

Functional equation

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

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: $\chi_{500} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 500,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7595545477\)
\(L(\frac12)\) \(\approx\) \(0.7595545477\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
good3 \( 1 - 1.61iT - T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 0.618iT - T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + 1.61iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.618iT - T^{2} \)
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

−11.44468091649843995632689383500, −10.23185777201554203323644435970, −9.865514844500552329516104641003, −8.738589451184618538346666106092, −8.412538737864736508454739740093, −6.90353233914736342663985772655, −5.84100056221284411154529286528, −4.99265802712363048191117049820, −4.24592654490052201150390067501, −3.07305886590573934492989252575, 1.06921259929186878510600224317, 2.25667130418092342522405004614, 3.49234463308560482312565558060, 4.91386951587758560842826163324, 6.17075501779488516761979415131, 7.17943912351348497531709641840, 8.022520811785433525116451494462, 8.827412960983563031709528189438, 10.01627556923669388752240569028, 10.87132219397361772674096076983

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