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Properties

Label 2-1488-31.5-c1-0-24
Degree $2$
Conductor $1488$
Sign $-0.134 + 0.990i$
Analytic cond. $11.8817$
Root an. cond. $3.44698$
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  + (0.5 − 0.866i)3-s + (1.81 − 3.14i)7-s + (−0.499 − 0.866i)9-s + (2.81 + 4.87i)11-s + (−3.10 − 5.37i)13-s + (0.784 − 1.35i)17-s + (1.31 − 2.28i)19-s + (−1.81 − 3.14i)21-s − 7.69·23-s + (2.5 − 4.33i)25-s − 0.999·27-s + 1.93·29-s + (5.38 + 1.41i)31-s + 5.63·33-s + (0.251 − 0.435i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.686 − 1.18i)7-s + (−0.166 − 0.288i)9-s + (0.849 + 1.47i)11-s + (−0.860 − 1.48i)13-s + (0.190 − 0.329i)17-s + (0.302 − 0.523i)19-s + (−0.396 − 0.686i)21-s − 1.60·23-s + (0.5 − 0.866i)25-s − 0.192·27-s + 0.359·29-s + (0.967 + 0.254i)31-s + 0.980·33-s + (0.0413 − 0.0716i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1488\)    =    \(2^{4} \cdot 3 \cdot 31\)
Sign: $-0.134 + 0.990i$
Analytic conductor: \(11.8817\)
Root analytic conductor: \(3.44698\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1488} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1488,\ (\ :1/2),\ -0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.869147927\)
\(L(\frac12)\) \(\approx\) \(1.869147927\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-5.38 - 1.41i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.81 + 3.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.81 - 4.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.10 + 5.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.784 + 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.31 + 2.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 - 1.93T + 29T^{2} \)
37 \( 1 + (-0.251 + 0.435i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.215 - 0.373i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.10 - 5.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.69T + 47T^{2} \)
53 \( 1 + (3.66 + 6.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.81 + 3.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-2.78 - 4.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.20 + 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.10 + 12.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.38 - 5.86i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 2.13T + 97T^{2} \)
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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

−9.379931480447398068078266960704, −8.169930890654578236881026632257, −7.65312769614127035501213954331, −7.08154783772050207604427934644, −6.17912164284588061006140819934, −4.83616790981141956320913726448, −4.35243960423894917031127071365, −3.07508576200113806689331380181, −1.93191274502611734642388472629, −0.73801368599126151457484806809, 1.60819247038392484824789708174, 2.65534876959772002674805239587, 3.78697987839384712194129538175, 4.60774751182490302301671648310, 5.66957370099382570470029286271, 6.21154479800270878825405013744, 7.42404422728442583064251334544, 8.427805192150267443244481076014, 8.840047254946715225061308635096, 9.525291660216280788313704088101

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