Properties

Label 2-300-25.6-c1-0-2
Degree $2$
Conductor $300$
Sign $0.268 + 0.963i$
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  + (−0.809 + 0.587i)3-s + (−0.233 − 2.22i)5-s − 0.511·7-s + (0.309 − 0.951i)9-s + (−0.564 − 1.73i)11-s + (1.89 − 5.82i)13-s + (1.49 + 1.66i)15-s + (−2.09 − 1.51i)17-s + (3.93 + 2.85i)19-s + (0.413 − 0.300i)21-s + (−2.04 − 6.30i)23-s + (−4.89 + 1.03i)25-s + (0.309 + 0.951i)27-s + (6.46 − 4.70i)29-s + (3.95 + 2.87i)31-s + ⋯
L(s)  = 1  + (−0.467 + 0.339i)3-s + (−0.104 − 0.994i)5-s − 0.193·7-s + (0.103 − 0.317i)9-s + (−0.170 − 0.523i)11-s + (0.524 − 1.61i)13-s + (0.386 + 0.429i)15-s + (−0.506 − 0.368i)17-s + (0.902 + 0.655i)19-s + (0.0902 − 0.0655i)21-s + (−0.427 − 1.31i)23-s + (−0.978 + 0.207i)25-s + (0.0594 + 0.183i)27-s + (1.20 − 0.872i)29-s + (0.709 + 0.515i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769674 - 0.584214i\)
\(L(\frac12)\) \(\approx\) \(0.769674 - 0.584214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.233 + 2.22i)T \)
good7 \( 1 + 0.511T + 7T^{2} \)
11 \( 1 + (0.564 + 1.73i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.89 + 5.82i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.09 + 1.51i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.93 - 2.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.04 + 6.30i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.46 + 4.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.95 - 2.87i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.29 - 7.07i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.77 + 5.45i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 + (6.43 - 4.67i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.07 - 0.781i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.11 - 9.59i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.47 - 4.53i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.5 - 8.42i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.34 + 3.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.07 + 3.31i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.06 + 0.775i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.738 + 0.536i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.63 - 11.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.98 + 4.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.65433852577961155296382938872, −10.50594358245174799362409332927, −9.844727378185704136446063348574, −8.565815723621145625708527056460, −8.001220342291190429718870629493, −6.37673206395480596582051034874, −5.46829990600871094692024157190, −4.51601674762235185969225233901, −3.12263568842217983236342604030, −0.78480386283946699059256817189, 1.95811561132110517878834212113, 3.53064220823209389421307818106, 4.87526299176482912353337160312, 6.35230509632815489347479165718, 6.85896162282145548011922845741, 7.88713631229999544244555587600, 9.269431409535875358463239327812, 10.12352669115360030591726623058, 11.33071583988933681774634430728, 11.57291994912658516237628687257

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