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Properties

Label 2-1014-13.3-c1-0-13
Degree $2$
Conductor $1014$
Sign $-0.434 + 0.900i$
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 − 4.29·5-s + (0.499 − 0.866i)6-s + (−2.17 + 3.77i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (2.14 + 3.72i)10-s + (0.579 + 1.00i)11-s − 0.999·12-s + 4.35·14-s + (−2.14 − 3.72i)15-s + (−0.5 − 0.866i)16-s + (−0.246 + 0.427i)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 − 1.92·5-s + (0.204 − 0.353i)6-s + (−0.823 + 1.42i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.679 + 1.17i)10-s + (0.174 + 0.302i)11-s − 0.288·12-s + 1.16·14-s + (−0.554 − 0.960i)15-s + (−0.125 − 0.216i)16-s + (−0.0599 + 0.103i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2061411602\)
\(L(\frac12)\) \(\approx\) \(0.2061411602\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.29T + 5T^{2} \)
7 \( 1 + (2.17 - 3.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.579 - 1.00i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.246 - 0.427i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.890 + 1.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.69 + 2.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.46 + 6.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + (1.93 + 3.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.15 + 5.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.69 - 6.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.78T + 47T^{2} \)
53 \( 1 + 2.51T + 53T^{2} \)
59 \( 1 + (-3.31 + 5.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.24 - 9.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.04 - 3.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.69 + 8.12i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.374T + 73T^{2} \)
79 \( 1 - 2.65T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (-0.417 - 0.723i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.19 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.563524831787826210446355089210, −8.894132121191050109076998626630, −8.268744032381804481604991212142, −7.49052995848728253876262827467, −6.38404482255597157437510207213, −5.02575571225083420254270902509, −4.06763248043248389763342897086, −3.32339771591142537931410097265, −2.41827252117514906345455009607, −0.12807846103339600809303576829, 1.02636570438211121589459432256, 3.38490341502036340055665657734, 3.78977462414689420356080173028, 4.92525986785725684938282112902, 6.46992896139045901115491020097, 7.07245327104209549665052115807, 7.66494600267487008071889287551, 8.248671369982745709217540631200, 9.170952000494496709861182945781, 10.20735577979971375505950979297

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