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Properties

Label 2-2000-100.11-c0-0-1
Degree $2$
Conductor $2000$
Sign $-0.260 + 0.965i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
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  + (−0.809 − 0.587i)9-s + (−1.53 − 1.11i)13-s + (−0.363 − 1.11i)17-s + (0.5 − 1.53i)29-s + (−0.951 − 0.690i)37-s + (0.5 + 0.363i)41-s + 49-s + (−0.587 + 1.80i)53-s + (0.5 − 0.363i)61-s + (1.53 − 1.11i)73-s + (0.309 + 0.951i)81-s + (−1.30 + 0.951i)89-s + (0.363 − 1.11i)97-s − 1.61·101-s + (−0.5 − 0.363i)109-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)9-s + (−1.53 − 1.11i)13-s + (−0.363 − 1.11i)17-s + (0.5 − 1.53i)29-s + (−0.951 − 0.690i)37-s + (0.5 + 0.363i)41-s + 49-s + (−0.587 + 1.80i)53-s + (0.5 − 0.363i)61-s + (1.53 − 1.11i)73-s + (0.309 + 0.951i)81-s + (−1.30 + 0.951i)89-s + (0.363 − 1.11i)97-s − 1.61·101-s + (−0.5 − 0.363i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $-0.260 + 0.965i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2000} (1551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2000,\ (\ :0),\ -0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7333702748\)
\(L(\frac12)\) \(\approx\) \(0.7333702748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.282781844669046413020592985526, −8.279447974189344340513696335779, −7.59447837400212752971619439322, −6.83023313490977931125416196919, −5.83798866795066162637890334076, −5.18179388072365709717245653532, −4.25401328979997746556322956999, −3.00869180905099925723995394103, −2.43648941956941320723457294503, −0.49435598869371392984368206503, 1.81641116715361197686119114472, 2.66956096770201140043368430675, 3.84127804581096523936710819011, 4.84387293093768339823996376621, 5.43476147540392267711147679789, 6.58394229423926752364679036708, 7.12341296616466444636435157258, 8.148669551595433607630963401051, 8.728090917585129606897154175515, 9.528917179490181244347573444266

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