Properties

Label 2-10e3-40.19-c0-0-0
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯
L(s)  = 1  − 2-s + 4-s − 0.618·7-s − 8-s + 9-s − 1.61·11-s + 1.61·13-s + 0.618·14-s + 16-s − 18-s + 0.618·19-s + 1.61·22-s + 1.61·23-s − 1.61·26-s − 0.618·28-s − 32-s + 36-s − 0.618·37-s − 0.618·38-s + 0.618·41-s − 1.61·44-s − 1.61·46-s + 1.61·47-s − 0.618·49-s + 1.61·52-s − 0.618·53-s + 0.618·56-s + ⋯

Functional equation

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

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: $\chi_{1000} (499, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6767991991\)
\(L(\frac12)\) \(\approx\) \(0.6767991991\)
\(L(1)\) 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 \( 1 - T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 + 1.61T + T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 0.618T + T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.61T + T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
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

−10.21909375937087143863763296476, −9.336281646077878582184375078988, −8.606313541949706043374262076575, −7.68557501242738725718451301627, −7.03934487728620344713907360437, −6.12132479717912161917720685954, −5.15686831524762743437841247795, −3.62933963483017492191714838091, −2.67523889781754086218996491939, −1.18834119397106713934378917459, 1.18834119397106713934378917459, 2.67523889781754086218996491939, 3.62933963483017492191714838091, 5.15686831524762743437841247795, 6.12132479717912161917720685954, 7.03934487728620344713907360437, 7.68557501242738725718451301627, 8.606313541949706043374262076575, 9.336281646077878582184375078988, 10.21909375937087143863763296476

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