Properties

Label 2-150-25.21-c1-0-3
Degree $2$
Conductor $150$
Sign $0.535 + 0.844i$
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 + (0.690 − 2.12i)5-s + (0.809 − 0.587i)6-s + 0.381·7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (−1.80 − 1.31i)10-s + (0.427 − 1.31i)11-s + (−0.309 − 0.951i)12-s + (0.763 + 2.35i)13-s + (0.118 − 0.363i)14-s + (1.80 − 1.31i)15-s + (0.309 + 0.951i)16-s + (2.61 − 1.90i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.309 − 0.951i)5-s + (0.330 − 0.239i)6-s + 0.144·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.572 − 0.415i)10-s + (0.128 − 0.396i)11-s + (−0.0892 − 0.274i)12-s + (0.211 + 0.652i)13-s + (0.0315 − 0.0970i)14-s + (0.467 − 0.339i)15-s + (0.0772 + 0.237i)16-s + (0.634 − 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
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: \((2,\ 150,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23320 - 0.677961i\)
\(L(\frac12)\) \(\approx\) \(1.23320 - 0.677961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.690 + 2.12i)T \)
good7 \( 1 - 0.381T + 7T^{2} \)
11 \( 1 + (-0.427 + 1.31i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.763 - 2.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.61 + 1.90i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.23 - 4.53i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.38 - 4.25i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.381 - 0.277i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.54 - 2.57i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.47 - 7.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.38 + 7.33i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (-9.47 - 6.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (7.35 + 5.34i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.427 + 1.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.23 + 6.88i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.47 + 6.15i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (11.7 + 8.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.85 + 11.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.73 - 1.98i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.16 - 5.20i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.85 + 11.8i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-4.54 - 3.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.80564295324049036716903454592, −11.98215083339237061554415778453, −10.82748300933680381712565068059, −9.734748605561883321750643071433, −8.937730417884018014575819450147, −7.977295075985077516107503227531, −6.03746520332227722150304872891, −4.78863267738919922204329203112, −3.64375192631268278900769211989, −1.77705325616513407938804107169, 2.54035898568887869145233121268, 4.09191407233855373180184861887, 5.82372521720221727346789161515, 6.79730757470107264759183094159, 7.76082225247954562434367029336, 8.829523272565290399427249198699, 10.08189558369387516896171485341, 11.09031684263398783549593562596, 12.56900383217744894988950868198, 13.26121827704303218769370158445

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