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Properties

Label 2-1500-25.16-c1-0-15
Degree $2$
Conductor $1500$
Sign $-0.813 + 0.581i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (0.309 + 0.951i)3-s − 1.31·7-s + (−0.809 + 0.587i)9-s + (−1.25 − 0.913i)11-s + (−1.96 + 1.42i)13-s + (−0.406 + 1.25i)17-s + (−0.315 + 0.971i)19-s + (−0.407 − 1.25i)21-s + (−4.05 − 2.94i)23-s + (−0.809 − 0.587i)27-s + (−1.82 − 5.61i)29-s + (2.73 − 8.41i)31-s + (0.480 − 1.47i)33-s + (−4.06 + 2.95i)37-s + (−1.96 − 1.42i)39-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s − 0.498·7-s + (−0.269 + 0.195i)9-s + (−0.379 − 0.275i)11-s + (−0.544 + 0.395i)13-s + (−0.0985 + 0.303i)17-s + (−0.0723 + 0.222i)19-s + (−0.0889 − 0.273i)21-s + (−0.844 − 0.613i)23-s + (−0.155 − 0.113i)27-s + (−0.338 − 1.04i)29-s + (0.491 − 1.51i)31-s + (0.0836 − 0.257i)33-s + (−0.668 + 0.485i)37-s + (−0.314 − 0.228i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08505653202\)
\(L(\frac12)\) \(\approx\) \(0.08505653202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
good7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + (1.25 + 0.913i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.96 - 1.42i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.406 - 1.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.315 - 0.971i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.05 + 2.94i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.82 + 5.61i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.73 + 8.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.06 - 2.95i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.43 - 4.67i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.84T + 43T^{2} \)
47 \( 1 + (-2.39 - 7.37i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.22 + 3.75i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.35 + 4.61i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.83 - 2.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.57 - 7.92i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.00 + 12.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.2 + 7.47i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.386 + 1.18i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.53 + 7.80i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.74 + 5.62i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.02 - 15.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.307197637325230341814919107955, −8.328179236754006371334507384575, −7.76607547440531773405169404721, −6.57717993067446193644841453888, −5.93895785580211628897287612536, −4.85135413599847329091389150839, −4.06442556998377266927693806470, −3.07566108312375420001211988666, −2.04268200127448765059205567524, −0.03041686967965401276679575008, 1.64849121574166987047177985285, 2.78671942640595886815628900873, 3.64503904410209988062033069800, 4.97400470096352289863522243496, 5.65908205204258018650797439637, 6.87940796967434006825354336162, 7.17185842008584039001582642822, 8.255407246749400625388092428112, 8.871200287363404153556816715747, 9.897412755009004662549825270955

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