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Properties

Label 2-10e3-8.3-c0-0-2
Degree $2$
Conductor $1000$
Sign $1$
Analytic cond. $0.499065$
Root an. cond. $0.706445$
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 − 4-s − 1.61i·7-s − i·8-s − 9-s + 0.618·11-s − 0.618i·13-s + 1.61·14-s + 16-s − i·18-s + 1.61·19-s + 0.618i·22-s − 0.618i·23-s + 0.618·26-s + 1.61i·28-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 1.61i·7-s − i·8-s − 9-s + 0.618·11-s − 0.618i·13-s + 1.61·14-s + 16-s − i·18-s + 1.61·19-s + 0.618i·22-s − 0.618i·23-s + 0.618·26-s + 1.61i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(0.499065\)
Root analytic conductor: \(0.706445\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8505397702\)
\(L(\frac12)\) \(\approx\) \(0.8505397702\)
\(L(1)\) 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 \( 1 + T^{2} \)
7 \( 1 + 1.61iT - T^{2} \)
11 \( 1 - 0.618T + T^{2} \)
13 \( 1 + 0.618iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
23 \( 1 + 0.618iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61iT - T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 0.618iT - T^{2} \)
53 \( 1 - 1.61iT - T^{2} \)
59 \( 1 - 1.61T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
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

−10.05140006647898626354693581657, −9.230219139527150670788931860659, −8.339414635898990353612887641657, −7.52763351210575740728574484470, −6.96030708153170355150733223390, −5.95425057796081598403791478785, −5.11089164664135601252674248407, −4.05044386761420648592790285583, −3.23438279179993723560700966856, −0.886336849121244270002008400291, 1.69926707956014038299708096864, 2.80119684492478915363972911188, 3.57312699061533403071699906158, 5.08697837931114319036111818488, 5.52315502847623226942703368889, 6.64651576831471675882014119021, 8.142710175677256728199506049288, 8.725658628376628377253661881080, 9.397382726591370986549255208385, 10.02137939848365779774768363054

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