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Properties

Label 2-3000-120.83-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7351898214\)
\(L(\frac12)\) \(\approx\) \(0.7351898214\)
\(L(1)\) 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

−8.884953980193077585594673795077, −8.253784955162389306765440565923, −7.29342437926365302359012004344, −6.57588650610513232162627667430, −5.89420077203523276337459854008, −5.27415117160562716649465563345, −4.31662784059314622917664128536, −3.94503554763126344709919605129, −2.61765860455199710730710007309, −0.53452421486130079287224393577, 1.24315813780761323139749738347, 1.98328577559691015743287289669, 3.18425297650594233280018557085, 4.03352682482588481060001680601, 5.09308638172566405936413247834, 5.66496599892148537191515532568, 6.37485100026037278055600825219, 7.54011997417212356811724771397, 8.025019381280907618292035684009, 8.937658942818684339571617439376

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