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Properties

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

Origins

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Normalization:  

Dirichlet series

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

\(L(\frac{1}{2})\) \(\approx\) \(0.8220717437\)
\(L(\frac12)\) \(\approx\) \(0.8220717437\)
\(L(1)\) 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 \( 1 + 0.618iT - T^{2} \)
7 \( 1 + 1.61iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.61iT - T^{2} \)
29 \( 1 + 0.618T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + 0.618iT - T^{2} \)
47 \( 1 - 0.618iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.61iT - T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
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

−11.00932626690238060210699390275, −10.07346951200950418173798424667, −9.348832613678366874757574880527, −8.006120872951264265066786573584, −7.37778346034146958242057896319, −6.88949502527873404401751038917, −5.75840755322082570063539996459, −4.48356678306683325587443462193, −3.66735397126259653442367922964, −1.25276422819773355308339692320, 2.00692432004278416078375153215, 3.10704401620495720348084907954, 4.37510281737693269411401181871, 5.18998964462435297827539165020, 6.26445790760845398235712485742, 7.903753242028869966867672037200, 8.930074537195376848494709426890, 9.367335642240957382684678719080, 10.34004485497887229454835413743, 11.05528460054153254276676768603

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