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Properties

Label 2-1014-13.4-c1-0-13
Degree $2$
Conductor $1014$
Sign $0.967 - 0.252i$
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  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − 2i·5-s + (−0.866 + 0.499i)6-s + (1.73 − i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)10-s − 0.999·12-s + 1.99·14-s + (1.73 + i)15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s − 0.999i·18-s + (5.19 − 3i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s − 0.894i·5-s + (−0.353 + 0.204i)6-s + (0.654 − 0.377i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s − 0.288·12-s + 0.534·14-s + (0.447 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s − 0.235i·18-s + (1.19 − 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.279569429\)
\(L(\frac12)\) \(\approx\) \(2.279569429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.19 + 3i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.66 - 5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3.46 - 2i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 6i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.811527441035148405556636213515, −9.298406871083625710700546975268, −8.013321247799395139194477854267, −7.72544790606469018359786050133, −6.29624559472414710370640791856, −5.59158091462670474381159881405, −4.59716699201408474500974769669, −4.23209384963344659878017277029, −2.81503036708817073237438680705, −1.08959055642862441260669015721, 1.33176286387248386211811266400, 2.58199019299838039445052739756, 3.42522957090486783690852490411, 4.82890356872882950882885569913, 5.51761029466553066029773442556, 6.53091056816621393154791539079, 7.19742763916180949464390153472, 8.108403177737003654483180070521, 9.147644219344242406015148913394, 10.33001153986386496030790063752

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