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Properties

Label 2-300-3.2-c8-0-12
Degree $2$
Conductor $300$
Sign $0.629 - 0.776i$
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  + (51.0 − 62.9i)3-s + 459.·7-s + (−1.35e3 − 6.41e3i)9-s − 9.13e3i·11-s − 4.80e4·13-s + 5.33e4i·17-s − 1.22e5·19-s + (2.34e4 − 2.88e4i)21-s + 2.63e5i·23-s + (−4.72e5 − 2.42e5i)27-s + 1.04e6i·29-s + 1.39e6·31-s + (−5.74e5 − 4.66e5i)33-s + 2.84e6·37-s + (−2.45e6 + 3.02e6i)39-s + ⋯
L(s)  = 1  + (0.629 − 0.776i)3-s + 0.191·7-s + (−0.206 − 0.978i)9-s − 0.624i·11-s − 1.68·13-s + 0.638i·17-s − 0.938·19-s + (0.120 − 0.148i)21-s + 0.943i·23-s + (−0.889 − 0.456i)27-s + 1.47i·29-s + 1.51·31-s + (−0.484 − 0.393i)33-s + 1.52·37-s + (−1.06 + 1.30i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.464943638\)
\(L(\frac12)\) \(\approx\) \(1.464943638\)
\(L(5)\) 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 \( 1 + (-51.0 + 62.9i)T \)
5 \( 1 \)
good7 \( 1 - 459.T + 5.76e6T^{2} \)
11 \( 1 + 9.13e3iT - 2.14e8T^{2} \)
13 \( 1 + 4.80e4T + 8.15e8T^{2} \)
17 \( 1 - 5.33e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.22e5T + 1.69e10T^{2} \)
23 \( 1 - 2.63e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.04e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.39e6T + 8.52e11T^{2} \)
37 \( 1 - 2.84e6T + 3.51e12T^{2} \)
41 \( 1 + 2.57e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.51e6T + 1.16e13T^{2} \)
47 \( 1 + 2.59e6iT - 2.38e13T^{2} \)
53 \( 1 - 2.74e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.78e7iT - 1.46e14T^{2} \)
61 \( 1 + 8.28e6T + 1.91e14T^{2} \)
67 \( 1 - 4.62e5T + 4.06e14T^{2} \)
71 \( 1 - 3.15e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.02e7T + 8.06e14T^{2} \)
79 \( 1 - 2.02e7T + 1.51e15T^{2} \)
83 \( 1 - 3.52e7iT - 2.25e15T^{2} \)
89 \( 1 + 4.31e7iT - 3.93e15T^{2} \)
97 \( 1 + 5.78e7T + 7.83e15T^{2} \)
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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

−10.40724907127916046206251416863, −9.384963913471618111182853099070, −8.471853288168272763371208421339, −7.65234923602302911264794727048, −6.77774514488522231399713568960, −5.70316553668554782513717903452, −4.37724771081187796650450825654, −3.07593196046743955485813683106, −2.15666504830220247114432252381, −0.989269967570179794557122510639, 0.28922435755204425537313193587, 2.16371138572784537352033959825, 2.81623219486229330563741722674, 4.48586623240911679438558877478, 4.72436328486191162268979855418, 6.32211694564525613850584148139, 7.59151217224496079467022791289, 8.272289636060678027258595405035, 9.598265879548434841017662884715, 9.866366281708231970054651938959

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