Properties

Label 2-150-25.11-c1-0-2
Degree $2$
Conductor $150$
Sign $0.573 + 0.819i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (2.17 + 0.506i)5-s + (−0.309 − 0.951i)6-s − 2.31·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (2.05 − 0.870i)10-s + (−2.77 + 2.01i)11-s + (−0.809 − 0.587i)12-s + (3.98 + 2.89i)13-s + (−1.86 + 1.35i)14-s + (1.15 − 1.91i)15-s + (−0.809 − 0.587i)16-s + (−2.36 − 7.29i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.973 + 0.226i)5-s + (−0.126 − 0.388i)6-s − 0.873·7-s + (−0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.651 − 0.275i)10-s + (−0.836 + 0.607i)11-s + (−0.233 − 0.169i)12-s + (1.10 + 0.803i)13-s + (−0.499 + 0.362i)14-s + (0.298 − 0.494i)15-s + (−0.202 − 0.146i)16-s + (−0.574 − 1.76i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.573 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42216 - 0.740344i\)
\(L(\frac12)\) \(\approx\) \(1.42216 - 0.740344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.506i)T \)
good7 \( 1 + 2.31T + 7T^{2} \)
11 \( 1 + (2.77 - 2.01i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.98 - 2.89i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.36 + 7.29i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1 - 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.61 - 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.13 - 6.56i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.09 - 3.37i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.20 + 2.32i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.132 - 0.0964i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.71T + 43T^{2} \)
47 \( 1 + (-1.92 + 5.93i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.69 + 11.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.62 + 6.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.249 + 0.181i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.42 + 4.39i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.117 + 0.362i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.0608 + 0.0442i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.72 + 14.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.403 - 1.24i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.58 - 2.60i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (5.31 - 16.3i)T + (-78.4 - 57.0i)T^{2} \)
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

−13.04765598506603660387063155612, −12.09123303792780870064958755906, −10.91635917048554370609795707288, −9.842747843412162036450805519288, −9.024234598634644245274655809722, −7.21840416858112163442932910279, −6.33911216518976143659001103226, −5.18756501823203964174365978041, −3.32682123668003054015689684564, −1.99240304180795183403286585906, 2.75874616185726000106929453455, 4.14198077624082652590633160123, 5.77741574399437854828444367026, 6.18735424333493060798999412893, 8.057675825684046333610144888687, 8.975850420138978963962986721864, 10.21648272276676334531789943509, 10.93566118005413828046244701476, 12.66906948019869678846209142256, 13.28092308461185613219878890310

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