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Properties

Label 2-3000-120.83-c0-0-7
Degree $2$
Conductor $3000$
Sign $-0.156 + 0.987i$
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  + (0.891 − 0.453i)2-s + (0.707 − 0.707i)3-s + (0.587 − 0.809i)4-s + (0.309 − 0.951i)6-s + (0.156 − 0.987i)8-s − 1.00i·9-s + (−0.156 − 0.987i)12-s + (−0.309 − 0.951i)16-s + (1.14 + 1.14i)17-s + (−0.453 − 0.891i)18-s + 0.618i·19-s + (−0.831 − 0.831i)23-s + (−0.587 − 0.809i)24-s + (−0.707 − 0.707i)27-s + 1.17i·31-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.707 − 0.707i)3-s + (0.587 − 0.809i)4-s + (0.309 − 0.951i)6-s + (0.156 − 0.987i)8-s − 1.00i·9-s + (−0.156 − 0.987i)12-s + (−0.309 − 0.951i)16-s + (1.14 + 1.14i)17-s + (−0.453 − 0.891i)18-s + 0.618i·19-s + (−0.831 − 0.831i)23-s + (−0.587 − 0.809i)24-s + (−0.707 − 0.707i)27-s + 1.17i·31-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3000\)    =    \(2^{3} \cdot 3 \cdot 5^{3}\)
Sign: $-0.156 + 0.987i$
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: \((2,\ 3000,\ (\ :0),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.689254527\)
\(L(\frac12)\) \(\approx\) \(2.689254527\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 + 0.453i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
19 \( 1 - 0.618iT - T^{2} \)
23 \( 1 + (0.831 + 0.831i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.17iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
53 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.90T + T^{2} \)
83 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
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

−8.492983256354153908733258332203, −8.004341240519717366490482681954, −7.08261705564599673406310149259, −6.31755814167346768598822239341, −5.77064229894849840534818780720, −4.68690580150282688541543080152, −3.70136155072711950395071124822, −3.17729361717719435002883016934, −2.07243769762540150934655307291, −1.29761134819807874483044021678, 1.97469728859097428775544188062, 2.98440302451253017930385855239, 3.56470048803015450737519225094, 4.50561676831862212314645338508, 5.14412091144028823143339149971, 5.88047555317088486697345942853, 6.88023291996442280700794576264, 7.84895416691842685415526191096, 7.956198087976874263889304132764, 9.216555426165302211057872334952

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