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Properties

Label 2-1014-13.9-c1-0-23
Degree $2$
Conductor $1014$
Sign $-0.990 - 0.134i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 2.13·5-s + (−0.499 − 0.866i)6-s + (−0.0244 − 0.0423i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.06 + 1.85i)10-s + (3.14 − 5.45i)11-s − 0.999·12-s − 0.0489·14-s + (−1.06 + 1.85i)15-s + (−0.5 + 0.866i)16-s + (1.44 + 2.50i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.955·5-s + (−0.204 − 0.353i)6-s + (−0.00924 − 0.0160i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.337 + 0.585i)10-s + (0.949 − 1.64i)11-s − 0.288·12-s − 0.0130·14-s + (−0.275 + 0.477i)15-s + (−0.125 + 0.216i)16-s + (0.350 + 0.607i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.990 - 0.134i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161446294\)
\(L(\frac12)\) \(\approx\) \(1.161446294\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 2.13T + 5T^{2} \)
7 \( 1 + (0.0244 + 0.0423i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.14 + 5.45i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.44 - 2.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.60 + 6.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.35 - 2.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.45 - 4.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 + (0.0881 - 0.152i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.29 - 7.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.35 + 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.20T + 47T^{2} \)
53 \( 1 - 9.34T + 53T^{2} \)
59 \( 1 + (2.13 + 3.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.55 + 6.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.69 + 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.35 + 7.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.9T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (-1.96 + 3.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.23 + 2.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.361884223350330692580462276775, −8.705676288825512239072287658961, −8.026694910859277286562516148177, −6.93090482708190216726784320357, −6.14214659458204118387614429504, −5.04869103132912338887535370086, −3.64996852849227812817929397529, −3.46427778106207052608353915587, −1.86821030736969631949843729010, −0.44741985832714980520524067479, 2.06493628089494876027843300420, 3.74007661928855701118812023176, 4.07071287945415083146305297097, 5.03915656967275562129261293691, 6.16950732738255903109788941889, 7.20652408657618803515130410204, 7.73417507662166089131767782665, 8.627823122695640906023782938203, 9.508440326301406095210207778094, 10.17998362539471517611033401387

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