Properties

Degree $2$
Conductor $1000$
Sign $-1$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + 0.618i·7-s i·8-s − 9-s − 1.61·11-s + 1.61i·13-s − 0.618·14-s + 16-s i·18-s − 0.618·19-s − 1.61i·22-s + 1.61i·23-s − 1.61·26-s − 0.618i·28-s + ⋯
L(s)  = 1  + i·2-s − 4-s + 0.618i·7-s i·8-s − 9-s − 1.61·11-s + 1.61i·13-s − 0.618·14-s + 16-s i·18-s − 0.618·19-s − 1.61i·22-s + 1.61i·23-s − 1.61·26-s − 0.618i·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: $-1$
Motivic weight: \(0\)
Character: $\chi_{1000} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5354403360\)
\(L(\frac12)\) \(\approx\) \(0.5354403360\)
\(L(1)\) 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 \( 1 - 0.618iT - T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 0.618T + T^{2} \)
23 \( 1 - 1.61iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 0.618iT - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.61iT - T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 + 0.618T + T^{2} \)
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 \( 1 - 1.61T + T^{2} \)
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

−10.43203885754029139844855445320, −9.446498523307904755142083271714, −8.810160839691377597651914720580, −8.092314340119241540856755082354, −7.27168070509826751643936382108, −6.27597403158564102147500296468, −5.51017068280432819385862507952, −4.82706710304148869006178771927, −3.55732012453145747468403924025, −2.24649263784152407964293391754, 0.48110794230330819991389489200, 2.50502165784129229413176958115, 3.04839952210451431850416556123, 4.35950311828659390244930121243, 5.27803604498195023920795062366, 6.03309163557421023811755445472, 7.67948355963812845910812034145, 8.146253669486179286464895710017, 8.972715820975204788441834558145, 10.13721911445573437365205234392

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