Properties

Label 2-150-15.8-c1-0-0
Degree $2$
Conductor $150$
Sign $-0.0706 - 0.997i$
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 + (−0.448 + 1.67i)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 + (−1.67 − 0.448i)12-s + 3.46·14-s − 1.00·16-s + (2.12 + 2.12i)17-s + (0.776 + 2.89i)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 + (−0.258 + 0.965i)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 + (−0.482 − 0.129i)12-s + 0.925·14-s − 0.250·16-s + (0.514 + 0.514i)17-s + (0.183 + 0.683i)18-s + 0.229i·19-s + (−0.654 − 1.13i)21-s + (0.783 − 0.783i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.0706 - 0.997i$
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: \((2,\ 150,\ (\ :1/2),\ -0.0706 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435683 + 0.467630i\)
\(L(\frac12)\) \(\approx\) \(0.435683 + 0.467630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.448 - 1.67i)T \)
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

−12.71153063187074083159210268185, −12.33131101584238971180483589677, −11.10656500890010521910426685002, −10.02328030831501633212113777622, −9.502427715299207933011712821255, −8.504845052830771884883383868091, −6.88221249942970180906677362897, −5.50210182948814418269338715932, −4.08571258127475767880558726187, −2.62999409128695463453044170967, 0.77580983440891378740683602000, 3.23893213817240536143298662573, 5.46264046827006235950537540646, 6.50716058750951685828092339165, 7.34294453234242202262700964100, 8.384613256730625838645691626742, 9.557690698564945787218760725381, 10.80036222117728419658783585575, 11.60954874753494182439645235430, 13.09881414294469434316503977616

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