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Properties

Label 2-150-25.16-c1-0-3
Degree $2$
Conductor $150$
Sign $0.992 + 0.125i$
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  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.690 − 2.12i)5-s + (0.309 − 0.951i)6-s + 2·7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.80 − 1.31i)10-s + (0.618 + 0.449i)11-s + (0.809 − 0.587i)12-s + (−1.5 + 1.08i)13-s + (1.61 + 1.17i)14-s − 2.23·15-s + (−0.809 + 0.587i)16-s + (0.354 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (0.309 − 0.951i)5-s + (0.126 − 0.388i)6-s + 0.755·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.572 − 0.415i)10-s + (0.186 + 0.135i)11-s + (0.233 − 0.169i)12-s + (−0.416 + 0.302i)13-s + (0.432 + 0.314i)14-s − 0.577·15-s + (−0.202 + 0.146i)16-s + (0.0858 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49453 - 0.0940282i\)
\(L(\frac12)\) \(\approx\) \(1.49453 - 0.0940282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.690 + 2.12i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-0.618 - 0.449i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.5 - 1.08i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.354 + 1.08i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.23 - 6.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.85 + 3.52i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.11 + 3.44i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3 - 9.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-7.16 + 5.20i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.11 - 2.99i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (-2.85 - 8.78i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.57 + 10.9i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.23 + 5.25i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.14 + 3.52i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.52 - 7.77i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.97 - 5.79i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.85 + 5.70i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.92 + 2.12i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.20 + 6.79i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.81728510557522293120648429568, −12.34290600349028552178431530951, −11.36033808836525419913030935056, −9.884079659220647862020962358029, −8.499241886291921564136946711727, −7.76726723566014285915545147718, −6.36060986634045218351650888427, −5.31749180948017034830304535510, −4.21501948897060548742146284126, −1.89414489831281707489994724004, 2.39424812705518774258246690486, 3.88651645387973616330895132183, 5.18463910985769388165492696502, 6.31836009160644899111225545213, 7.64982664406823497799484814159, 9.243439402267130846148126510136, 10.26938872061852166842472006831, 11.12197011188085210931820762365, 11.73524015727050511716928480188, 13.16615254079391933567342796186

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