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Properties

Label 2-300-3.2-c2-0-12
Degree $2$
Conductor $300$
Sign $-0.666 + 0.745i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2.23i)3-s − 8·7-s + (−1.00 − 8.94i)9-s − 8.94i·11-s − 12·13-s − 31.3i·17-s + 6·19-s + (−16 + 17.8i)21-s − 4.47i·23-s + (−22.0 − 15.6i)27-s + 26.8i·29-s + 34·31-s + (−20.0 − 17.8i)33-s − 44·37-s + (−24 + 26.8i)39-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)3-s − 1.14·7-s + (−0.111 − 0.993i)9-s − 0.813i·11-s − 0.923·13-s − 1.84i·17-s + 0.315·19-s + (−0.761 + 0.851i)21-s − 0.194i·23-s + (−0.814 − 0.579i)27-s + 0.925i·29-s + 1.09·31-s + (−0.606 − 0.542i)33-s − 1.18·37-s + (−0.615 + 0.688i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.666 + 0.745i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.535167 - 1.19667i\)
\(L(\frac12)\) \(\approx\) \(0.535167 - 1.19667i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2 + 2.23i)T \)
5 \( 1 \)
good7 \( 1 + 8T + 49T^{2} \)
11 \( 1 + 8.94iT - 121T^{2} \)
13 \( 1 + 12T + 169T^{2} \)
17 \( 1 + 31.3iT - 289T^{2} \)
19 \( 1 - 6T + 361T^{2} \)
23 \( 1 + 4.47iT - 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 - 34T + 961T^{2} \)
37 \( 1 + 44T + 1.36e3T^{2} \)
41 \( 1 - 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 28T + 1.84e3T^{2} \)
47 \( 1 - 4.47iT - 2.20e3T^{2} \)
53 \( 1 + 40.2iT - 2.80e3T^{2} \)
59 \( 1 + 98.3iT - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 - 92T + 4.48e3T^{2} \)
71 \( 1 - 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 56T + 5.32e3T^{2} \)
79 \( 1 - 78T + 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 - 32T + 9.40e3T^{2} \)
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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

−11.40335385784830607465475228971, −9.919435320945205768970169907586, −9.325249105445537376999411629286, −8.306418873260894727391541496925, −7.16441387574785811788768865694, −6.55938648755578187452958607733, −5.18529702695956408999013598437, −3.41167699065757616512106514696, −2.60169216266832697130234589791, −0.56330660833230031658104037037, 2.26416637353483593926308681575, 3.52706854472532155384578207998, 4.52760557085285635632207330829, 5.87456657182670705147906524786, 7.10634137949585791191249430813, 8.156892447225875542044634213066, 9.215912645048947635053377477771, 10.00816980617928662807735869112, 10.48923469525591158892951829158, 11.99558214838770257438958303892

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