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Properties

Label 2-3000-600.29-c0-0-1
Degree $2$
Conductor $3000$
Sign $-0.815 - 0.578i$
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  + (0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + 1.61i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.500 + 1.53i)14-s + (0.309 + 0.951i)16-s − i·18-s + (−1.30 − 0.951i)21-s + (−0.363 + 0.5i)22-s − 24-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)6-s + 1.61i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.190 + 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.500 + 1.53i)14-s + (0.309 + 0.951i)16-s − i·18-s + (−1.30 − 0.951i)21-s + (−0.363 + 0.5i)22-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3000\)    =    \(2^{3} \cdot 3 \cdot 5^{3}\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(1.49719\)
Root analytic conductor: \(1.22359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3000} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3000,\ (\ :0),\ -0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.741382408\)
\(L(\frac12)\) \(\approx\) \(1.741382408\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.90 - 0.618i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
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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

−9.295055156301850846239044460035, −8.420685668826290489575356655706, −7.65260877905248926044104020481, −6.57180799016103597969527139292, −5.91113629483319415675073612259, −5.44920687223336257653792936852, −4.66721895484871155346969741092, −3.93345327672200981512925577165, −2.86996796584863309987214594671, −2.10003122311460387177630029516, 0.848080293916129019972413991473, 1.78580265918927832326272359386, 3.07087341754234339060686095682, 3.85936740757604328308389646387, 4.81665527713586476835013186912, 5.44534901189422857572614896201, 6.42217833680091024759842642537, 6.91206320915450664059549230428, 7.55809956707375184924204596125, 8.296725819229072262623828811244

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