Properties

Label 2-150-75.53-c1-0-0
Degree $2$
Conductor $150$
Sign $-0.959 - 0.283i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
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  + (−0.891 + 0.453i)2-s + (−0.873 − 1.49i)3-s + (0.587 − 0.809i)4-s + (−2.21 − 0.314i)5-s + (1.45 + 0.935i)6-s + (−2.72 + 2.72i)7-s + (−0.156 + 0.987i)8-s + (−1.47 + 2.61i)9-s + (2.11 − 0.725i)10-s + (0.335 + 0.109i)11-s + (−1.72 − 0.172i)12-s + (−1.12 + 2.20i)13-s + (1.19 − 3.66i)14-s + (1.46 + 3.58i)15-s + (−0.309 − 0.951i)16-s + (−3.49 − 0.554i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (−0.504 − 0.863i)3-s + (0.293 − 0.404i)4-s + (−0.990 − 0.140i)5-s + (0.594 + 0.382i)6-s + (−1.03 + 1.03i)7-s + (−0.0553 + 0.349i)8-s + (−0.491 + 0.871i)9-s + (0.668 − 0.229i)10-s + (0.101 + 0.0328i)11-s + (−0.497 − 0.0497i)12-s + (−0.312 + 0.612i)13-s + (0.318 − 0.980i)14-s + (0.378 + 0.925i)15-s + (−0.0772 − 0.237i)16-s + (−0.848 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.959 - 0.283i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ -0.959 - 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00516079 + 0.0356955i\)
\(L(\frac12)\) \(\approx\) \(0.00516079 + 0.0356955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 + (0.873 + 1.49i)T \)
5 \( 1 + (2.21 + 0.314i)T \)
good7 \( 1 + (2.72 - 2.72i)T - 7iT^{2} \)
11 \( 1 + (-0.335 - 0.109i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.12 - 2.20i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (3.49 + 0.554i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.84 + 5.29i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.55 + 6.98i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-5.05 - 3.67i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.39 - 2.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.33 - 2.20i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (8.06 - 2.62i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.16 + 5.16i)T + 43iT^{2} \)
47 \( 1 + (-0.668 - 4.22i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-4.34 + 0.688i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (0.713 + 2.19i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.0451 + 0.139i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.18 + 7.47i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (3.62 - 4.98i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.30 - 4.74i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-0.803 + 1.10i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.915 - 5.77i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.633 - 1.94i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.93 + 1.09i)T + (92.2 - 29.9i)T^{2} \)
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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

−13.16929722261356010945458052107, −12.31934039717655028610061861193, −11.62556113522040522498672799398, −10.54676636384370567797403140653, −8.994084628573038683847443401089, −8.379955800811090219419488312687, −6.89218552401673325865973815457, −6.45066218345151852667107271785, −4.81082117728733546414437466973, −2.52971131558187216425992012726, 0.04313656448321838951638068651, 3.42254086959155153570432080165, 4.21767449343086224362576091322, 6.16242927844901037530510236312, 7.33865399323986936302272066368, 8.509739737251639518531568762332, 9.876269666257262664829326151053, 10.36844593385915770728056763058, 11.36603907284106736249301729883, 12.23814299108347309777515853513

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