Properties

Label 2-300-5.2-c8-0-15
Degree $2$
Conductor $300$
Sign $0.973 - 0.229i$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (33.0 − 33.0i)3-s + (2.96e3 + 2.96e3i)7-s − 2.18e3i·9-s + 2.70e4·11-s + (1.63e4 − 1.63e4i)13-s + (5.21e4 + 5.21e4i)17-s + 9.93e4i·19-s + 1.95e5·21-s + (2.52e5 − 2.52e5i)23-s + (−7.23e4 − 7.23e4i)27-s + 1.83e5i·29-s + 3.87e5·31-s + (8.93e5 − 8.93e5i)33-s + (−1.68e6 − 1.68e6i)37-s − 1.08e6i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.23 + 1.23i)7-s − 0.333i·9-s + 1.84·11-s + (0.573 − 0.573i)13-s + (0.624 + 0.624i)17-s + 0.762i·19-s + 1.00·21-s + (0.902 − 0.902i)23-s + (−0.136 − 0.136i)27-s + 0.259i·29-s + 0.420·31-s + (0.753 − 0.753i)33-s + (−0.899 − 0.899i)37-s − 0.468i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.063287535\)
\(L(\frac12)\) \(\approx\) \(4.063287535\)
\(L(5)\) 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 \( 1 + (-33.0 + 33.0i)T \)
5 \( 1 \)
good7 \( 1 + (-2.96e3 - 2.96e3i)T + 5.76e6iT^{2} \)
11 \( 1 - 2.70e4T + 2.14e8T^{2} \)
13 \( 1 + (-1.63e4 + 1.63e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-5.21e4 - 5.21e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 9.93e4iT - 1.69e10T^{2} \)
23 \( 1 + (-2.52e5 + 2.52e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.83e5iT - 5.00e11T^{2} \)
31 \( 1 - 3.87e5T + 8.52e11T^{2} \)
37 \( 1 + (1.68e6 + 1.68e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 1.14e6T + 7.98e12T^{2} \)
43 \( 1 + (2.09e6 - 2.09e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (2.73e4 + 2.73e4i)T + 2.38e13iT^{2} \)
53 \( 1 + (3.82e6 - 3.82e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.30e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.59e7T + 1.91e14T^{2} \)
67 \( 1 + (1.20e7 + 1.20e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 3.95e7T + 6.45e14T^{2} \)
73 \( 1 + (2.66e7 - 2.66e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 7.09e7iT - 1.51e15T^{2} \)
83 \( 1 + (-1.06e7 + 1.06e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 4.35e7iT - 3.93e15T^{2} \)
97 \( 1 + (5.13e6 + 5.13e6i)T + 7.83e15iT^{2} \)
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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

−10.42939599452702450111086769266, −8.997997400563770934073019772650, −8.649991803602470304580160525526, −7.69048724102959854014823510184, −6.40782698054831777960807939713, −5.59315530323118990341193516812, −4.29155192723700205847231917118, −3.11289960522390672742062189748, −1.77324047619569829880648011119, −1.14814352762267615776199555666, 0.952348276526806160289634325284, 1.63012541789907490049798585908, 3.39760461953417491560705904350, 4.20774002872111435756800659359, 5.04891215028663773948239071534, 6.66104036973691550822585008439, 7.40601454151955177369013553291, 8.532961683703872248417147692892, 9.300403685746531171895285103614, 10.28426297580861748892635213354

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