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Properties

Label 2-3000-120.83-c0-0-4
Degree $2$
Conductor $3000$
Sign $-0.891 + 0.453i$
Analytic cond. $1.49719$
Root an. cond. $1.22359$
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  + (−0.156 − 0.987i)2-s + (0.707 − 0.707i)3-s + (−0.951 + 0.309i)4-s + (−0.809 − 0.587i)6-s + (0.453 + 0.891i)8-s − 1.00i·9-s + (−0.453 + 0.891i)12-s + (0.809 − 0.587i)16-s + (−0.437 − 0.437i)17-s + (−0.987 + 0.156i)18-s − 1.61i·19-s + (1.34 + 1.34i)23-s + (0.951 + 0.309i)24-s + (−0.707 − 0.707i)27-s − 1.90i·31-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (0.707 − 0.707i)3-s + (−0.951 + 0.309i)4-s + (−0.809 − 0.587i)6-s + (0.453 + 0.891i)8-s − 1.00i·9-s + (−0.453 + 0.891i)12-s + (0.809 − 0.587i)16-s + (−0.437 − 0.437i)17-s + (−0.987 + 0.156i)18-s − 1.61i·19-s + (1.34 + 1.34i)23-s + (0.951 + 0.309i)24-s + (−0.707 − 0.707i)27-s − 1.90i·31-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3000\)    =    \(2^{3} \cdot 3 \cdot 5^{3}\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(1.49719\)
Root analytic conductor: \(1.22359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3000} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3000,\ (\ :0),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238622855\)
\(L(\frac12)\) \(\approx\) \(1.238622855\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.156 + 0.987i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
19 \( 1 + 1.61iT - T^{2} \)
23 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.90iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
53 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.17iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.17T + T^{2} \)
83 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.836965744708412286746222988080, −7.911672538443768421518584706143, −7.33390865574629412819938990104, −6.50555715392937822198351221577, −5.33153895149365422475474705842, −4.50194424109315458820191480996, −3.49067935759549958158787008376, −2.77463110028854884801726478209, −1.97684587860473898081114525085, −0.78663039336676680411010756623, 1.58887881654982909435850703792, 3.03039086503546177723656275994, 3.86169374170973125935725898949, 4.70111815830441370150393993797, 5.29731734359834411864668277817, 6.32697120457053372371846932645, 6.98122765666304280028167578127, 8.001840801213475232553319700852, 8.361027552076931357535336525668, 9.090078763436972611822277251330

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