Properties

Label 2-150-15.14-c2-0-10
Degree $2$
Conductor $150$
Sign $-0.997 + 0.0710i$
Analytic cond. $4.08720$
Root an. cond. $2.02168$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (−1.52 − 2.58i)3-s + 2.00·4-s + (2.16 + 3.65i)6-s − 7.48i·7-s − 2.82·8-s + (−4.32 + 7.89i)9-s − 8.48i·11-s + (−3.05 − 5.16i)12-s + 10i·13-s + 10.5i·14-s + 4.00·16-s − 30.3·17-s + (6.11 − 11.1i)18-s − 26.9·19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.509 − 0.860i)3-s + 0.500·4-s + (0.360 + 0.608i)6-s − 1.06i·7-s − 0.353·8-s + (−0.480 + 0.876i)9-s − 0.771i·11-s + (−0.254 − 0.430i)12-s + 0.769i·13-s + 0.756i·14-s + 0.250·16-s − 1.78·17-s + (0.339 − 0.620i)18-s − 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.997 + 0.0710i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ -0.997 + 0.0710i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0142472 - 0.400400i\)
\(L(\frac12)\) \(\approx\) \(0.0142472 - 0.400400i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (1.52 + 2.58i)T \)
5 \( 1 \)
good7 \( 1 + 7.48iT - 49T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 + 30.3T + 289T^{2} \)
19 \( 1 + 26.9T + 361T^{2} \)
23 \( 1 + 9.17T + 529T^{2} \)
29 \( 1 + 26.8iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 - 15.9iT - 1.36e3T^{2} \)
41 \( 1 + 47.3iT - 1.68e3T^{2} \)
43 \( 1 - 14.4iT - 1.84e3T^{2} \)
47 \( 1 - 45.8T + 2.20e3T^{2} \)
53 \( 1 - 30.3T + 2.80e3T^{2} \)
59 \( 1 + 24.0iT - 3.48e3T^{2} \)
61 \( 1 + 53.9T + 3.72e3T^{2} \)
67 \( 1 + 110. iT - 4.48e3T^{2} \)
71 \( 1 + 15.5iT - 5.04e3T^{2} \)
73 \( 1 + 87.9iT - 5.32e3T^{2} \)
79 \( 1 - 46.9T + 6.24e3T^{2} \)
83 \( 1 + 26.1T + 6.88e3T^{2} \)
89 \( 1 - 60.7iT - 7.92e3T^{2} \)
97 \( 1 - 36.0iT - 9.40e3T^{2} \)
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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

−12.12816809605445874752726013329, −11.06223002883698407515651777647, −10.59419257393530582340128351228, −9.005839799997872807197885678274, −8.032795575778598236699001028578, −6.88904809402180169818266718208, −6.21583623299624851868169856107, −4.31478866655728461635242819586, −2.09401247491402335706387530078, −0.32153733441475541077345720991, 2.46188029065458081133822226063, 4.37911463312378938018186272948, 5.70935628712746581444041980623, 6.78288712689685805278852600274, 8.489427075547043125339745179540, 9.136368566088944404949270678315, 10.26079715562368399392088670044, 11.02359085936710540531957428953, 12.05935988611673115633364343071, 12.92689117605290407376456347926

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