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Properties

Label 2-300-25.21-c1-0-0
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: $\chi_{300} (121, \cdot )$
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

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

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