Properties

Label 2-150-25.21-c1-0-0
Degree $2$
Conductor $150$
Sign $-0.967 + 0.254i$
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  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−2.12 + 0.697i)5-s + (0.809 − 0.587i)6-s − 4.25·7-s + (0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.00655 − 2.23i)10-s + (−1.51 + 4.64i)11-s + (0.309 + 0.951i)12-s + (−1.43 − 4.41i)13-s + (1.31 − 4.04i)14-s + (2.12 + 0.684i)15-s + (0.309 + 0.951i)16-s + (0.815 − 0.592i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.950 + 0.311i)5-s + (0.330 − 0.239i)6-s − 1.60·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.00207 − 0.707i)10-s + (−0.455 + 1.40i)11-s + (0.0892 + 0.274i)12-s + (−0.397 − 1.22i)13-s + (0.351 − 1.08i)14-s + (0.549 + 0.176i)15-s + (0.0772 + 0.237i)16-s + (0.197 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.967 + 0.254i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ -0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0155833 - 0.120511i\)
\(L(\frac12)\) \(\approx\) \(0.0155833 - 0.120511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (2.12 - 0.697i)T \)
good7 \( 1 + 4.25T + 7T^{2} \)
11 \( 1 + (1.51 - 4.64i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.43 + 4.41i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.815 + 0.592i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1 + 0.726i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.38 - 4.25i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.42 + 2.48i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.826 - 0.600i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.31 - 10.1i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.42 - 4.37i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-1.63 - 1.19i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.94 + 6.49i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.656 + 2.02i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.19 + 3.68i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.03 - 2.20i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.1 + 7.37i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.05 + 3.26i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.16 + 3.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.70 - 4.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.693 + 2.13i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.49 - 4.71i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.26887751539126525759213998348, −12.71222754954233316458842911208, −11.71748607743631580042553945873, −10.23483516602632224833834479116, −9.666726288904029199658937839545, −7.931867889108133038822271548954, −7.24646523512585555169545395311, −6.28335790982375308900983032028, −4.90525686306065385207265206139, −3.21643557336723267983573575044, 0.12975565448505981835450572316, 3.17168820940116161791242581814, 4.17813204650347352067016675566, 5.81986639519649156966693179357, 7.14737537220537566714983850100, 8.629911242685629168307768750569, 9.462287450953543365787961374398, 10.57456971507735571064990910840, 11.45587392234815584568446224878, 12.35197723116990013678804327734

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