↑

Properties

Label 2-150-5.2-c2-0-2
Degree $2$
Conductor $150$
Sign $0.850 + 0.525i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s − 2.44·6-s + (8.89 + 8.89i)7-s + (2 − 2i)8-s − 2.99i·9-s + 5.79·11-s + (2.44 + 2.44i)12-s + (6.79 − 6.79i)13-s − 17.7i·14-s − 4·16-s + (−6.10 − 6.10i)17-s + (−2.99 + 2.99i)18-s + 6.20i·19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s − 0.408·6-s + (1.27 + 1.27i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + 0.527·11-s + (0.204 + 0.204i)12-s + (0.522 − 0.522i)13-s − 1.27i·14-s − 0.250·16-s + (−0.358 − 0.358i)17-s + (−0.166 + 0.166i)18-s + 0.326i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.41340 - 0.401517i\)
\(L(\frac12)\) \(\approx\) \(1.41340 - 0.401517i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-8.89 - 8.89i)T + 49iT^{2} \)
11 \( 1 - 5.79T + 121T^{2} \)
13 \( 1 + (-6.79 + 6.79i)T - 169iT^{2} \)
17 \( 1 + (6.10 + 6.10i)T + 289iT^{2} \)
19 \( 1 - 6.20iT - 361T^{2} \)
23 \( 1 + (-18.6 + 18.6i)T - 529iT^{2} \)
29 \( 1 + 6.20iT - 841T^{2} \)
31 \( 1 + 0.404T + 961T^{2} \)
37 \( 1 + (-27 - 27i)T + 1.36e3iT^{2} \)
41 \( 1 + 1.79T + 1.68e3T^{2} \)
43 \( 1 + (36.4 - 36.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (38.6 + 38.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (69.0 - 69.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 20iT - 3.48e3T^{2} \)
61 \( 1 + 63.1T + 3.72e3T^{2} \)
67 \( 1 + (40.0 + 40.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 25.7T + 5.04e3T^{2} \)
73 \( 1 + (-56.7 + 56.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 139. iT - 6.24e3T^{2} \)
83 \( 1 + (13.7 - 13.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 58.6iT - 7.92e3T^{2} \)
97 \( 1 + (-15.9 - 15.9i)T + 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.45775296367321021553821856383, −11.68198221840823114197831264580, −10.85617721888967029012895231453, −9.367522145788767262890569705682, −8.557574506489458834307738818956, −7.84267456549198226663332823195, −6.28460750206279662014748992188, −4.76255071357263434479267752206, −2.90200222253351218130086710765, −1.55195650699170114848274522903, 1.47392996870541241635829418045, 3.91327237702376019342762608077, 4.99099431593495592982035325190, 6.70564305628609624950958537734, 7.71321314869355444020023296110, 8.638300960339670238699546655536, 9.661780822391235709839237523421, 10.87115299188581069024312786617, 11.36455013997043282885875530387, 13.23812583798002085699614234722

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