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Properties

Label 2-150-15.2-c1-0-5
Degree $2$
Conductor $150$
Sign $0.0618 + 0.998i$
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 + (0.292 − 1.70i)3-s − 1.00i·4-s + (−0.999 − 1.41i)6-s + (1 + i)7-s + (−0.707 − 0.707i)8-s + (−2.82 − i)9-s + 1.41i·11-s + (−1.70 − 0.292i)12-s + 1.41·14-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−2.70 + 1.29i)18-s + 4i·19-s + (2 − 1.41i)21-s + (1.00 + 1.00i)22-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.169 − 0.985i)3-s − 0.500i·4-s + (−0.408 − 0.577i)6-s + (0.377 + 0.377i)7-s + (−0.250 − 0.250i)8-s + (−0.942 − 0.333i)9-s + 0.426i·11-s + (−0.492 − 0.0845i)12-s + 0.377·14-s − 0.250·16-s + (0.342 − 0.342i)17-s + (−0.638 + 0.304i)18-s + 0.917i·19-s + (0.436 − 0.308i)21-s + (0.213 + 0.213i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.0618 + 0.998i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07291 - 1.00846i\)
\(L(\frac12)\) \(\approx\) \(1.07291 - 1.00846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 \)
good7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (6 + 6i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (2.82 + 2.82i)T + 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-4 - 4i)T + 67iT^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−12.62801705934496964330163603216, −12.03345843458782057860913525056, −11.10201394673275814285660561685, −9.777141402896229544445029296531, −8.549842992853152009203692468559, −7.43143313213841432231629025837, −6.20138554022249356674161500386, −5.00830344452743109570541413242, −3.19366268773046050282275955484, −1.69579203176320333609197701026, 3.07132677816928570787902344062, 4.38952672316284747988970119318, 5.34819308263600551574721031375, 6.73482976201572640241325291537, 8.142243146865085970514718100456, 8.985692380732752015401268507962, 10.32989234836431187015106874474, 11.15083634974028438125166343030, 12.29516435834710054854808524304, 13.64689751035716128818865256128

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