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Properties

Label 2-150-15.8-c1-0-4
Degree $2$
Conductor $150$
Sign $0.828 - 0.559i$
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  + (0.707 + 0.707i)2-s + (1.67 − 0.448i)3-s + 1.00i·4-s + (1.5 + 0.866i)6-s + (−2.44 + 2.44i)7-s + (−0.707 + 0.707i)8-s + (2.59 − 1.50i)9-s − 5.19i·11-s + (0.448 + 1.67i)12-s − 3.46·14-s − 1.00·16-s + (−2.12 − 2.12i)17-s + (2.89 + 0.776i)18-s + i·19-s + (−3 + 5.19i)21-s + (3.67 − 3.67i)22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.965 − 0.258i)3-s + 0.500i·4-s + (0.612 + 0.353i)6-s + (−0.925 + 0.925i)7-s + (−0.250 + 0.250i)8-s + (0.866 − 0.5i)9-s − 1.56i·11-s + (0.129 + 0.482i)12-s − 0.925·14-s − 0.250·16-s + (−0.514 − 0.514i)17-s + (0.683 + 0.183i)18-s + 0.229i·19-s + (−0.654 + 1.13i)21-s + (0.783 − 0.783i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.828 - 0.559i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.828 - 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63019 + 0.499193i\)
\(L(\frac12)\) \(\approx\) \(1.63019 + 0.499193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
5 \( 1 \)
good7 \( 1 + (2.44 - 2.44i)T - 7iT^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (2.12 + 2.12i)T + 17iT^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (4.24 - 4.24i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (-2.44 - 2.44i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (3.67 - 3.67i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-6.12 - 6.12i)T + 73iT^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + (-2.12 + 2.12i)T - 83iT^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−13.33061749048803286923780687149, −12.45726626675015378193100454794, −11.36671939821205130573798369171, −9.668539346855621596939464055384, −8.833714664259009541260787522317, −7.927639899796082279395426842762, −6.59019295305139983385951764781, −5.65662929676451807619200184444, −3.75160830104294674073228019702, −2.70572854985074972695335552668, 2.21477996751888029364567173423, 3.74289328770649006938144316249, 4.59037963425345594477193943323, 6.57907450940759323929561227870, 7.57105743245793777515742952631, 9.082740597853566796540835246490, 10.02898589089507563784961585740, 10.58104763866450892990411159604, 12.27648274533574283403614565765, 13.00216077312138081701779265833

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