Properties

Label 2-300-25.4-c1-0-5
Degree $2$
Conductor $300$
Sign $0.275 + 0.961i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)3-s + (1.28 − 1.82i)5-s − 2.44i·7-s + (−0.309 − 0.951i)9-s + (−0.178 + 0.548i)11-s + (−6.13 + 1.99i)13-s + (−0.720 − 2.11i)15-s + (1.11 + 1.53i)17-s + (6.69 − 4.86i)19-s + (−1.97 − 1.43i)21-s + (4.00 + 1.30i)23-s + (−1.67 − 4.70i)25-s + (−0.951 − 0.309i)27-s + (5.28 + 3.84i)29-s + (−3.93 + 2.86i)31-s + ⋯
L(s)  = 1  + (0.339 − 0.467i)3-s + (0.576 − 0.817i)5-s − 0.923i·7-s + (−0.103 − 0.317i)9-s + (−0.0537 + 0.165i)11-s + (−1.70 + 0.552i)13-s + (−0.186 − 0.546i)15-s + (0.270 + 0.372i)17-s + (1.53 − 1.11i)19-s + (−0.431 − 0.313i)21-s + (0.834 + 0.271i)23-s + (−0.335 − 0.941i)25-s + (−0.183 − 0.0594i)27-s + (0.982 + 0.713i)29-s + (−0.707 + 0.513i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19177 - 0.897908i\)
\(L(\frac12)\) \(\approx\) \(1.19177 - 0.897908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (-1.28 + 1.82i)T \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + (0.178 - 0.548i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (6.13 - 1.99i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.11 - 1.53i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.69 + 4.86i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.00 - 1.30i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.28 - 3.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.93 - 2.86i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.207 - 0.0673i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.99 - 6.13i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.42iT - 43T^{2} \)
47 \( 1 + (-5.65 + 7.78i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.22 - 11.3i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.72 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.48 - 4.57i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.27 - 3.13i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.24 - 3.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.55 + 3.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.83 + 2.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.01 + 2.77i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.75 + 5.40i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.87 - 5.32i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.80332875888928092572147940005, −10.44591458396675689751498311335, −9.557861197224605771270113279684, −8.850494986082860478869992581362, −7.46174283373205198494040946938, −6.99688776956888986718504945599, −5.38564175360711896301670933128, −4.49371309387433637108858582259, −2.79110941649770992519751152831, −1.18872030815045694783505217814, 2.39003077894925875365035859364, 3.23789305193134631745631565273, 5.04849783449001651621907472517, 5.79938745424641265760261021282, 7.18812766194128153829792279566, 8.076250303129559710543151938400, 9.557280366512215567220861041621, 9.741159646631232885540809021948, 10.89516377220426915751748532777, 11.92948401397701443201917571968

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