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Properties

Label 2-1071-7.4-c1-0-49
Degree $2$
Conductor $1071$
Sign $-0.815 - 0.579i$
Analytic cond. $8.55197$
Root an. cond. $2.92437$
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  + (−0.614 − 1.06i)2-s + (0.243 − 0.422i)4-s + (−1.20 − 2.09i)5-s + (−1.25 − 2.33i)7-s − 3.05·8-s + (−1.48 + 2.57i)10-s + (1.73 − 3.01i)11-s + 5.95·13-s + (−1.71 + 2.76i)14-s + (1.39 + 2.41i)16-s + (0.5 − 0.866i)17-s + (−3.59 − 6.21i)19-s − 1.18·20-s − 4.27·22-s + (−1.40 − 2.43i)23-s + ⋯
L(s)  = 1  + (−0.434 − 0.753i)2-s + (0.121 − 0.211i)4-s + (−0.541 − 0.937i)5-s + (−0.473 − 0.881i)7-s − 1.08·8-s + (−0.470 + 0.814i)10-s + (0.524 − 0.908i)11-s + 1.65·13-s + (−0.457 + 0.739i)14-s + (0.348 + 0.603i)16-s + (0.121 − 0.210i)17-s + (−0.823 − 1.42i)19-s − 0.263·20-s − 0.912·22-s + (−0.292 − 0.507i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1071\)    =    \(3^{2} \cdot 7 \cdot 17\)
Sign: $-0.815 - 0.579i$
Analytic conductor: \(8.55197\)
Root analytic conductor: \(2.92437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1071} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1071,\ (\ :1/2),\ -0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9610937382\)
\(L(\frac12)\) \(\approx\) \(0.9610937382\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (1.25 + 2.33i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.614 + 1.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.20 + 2.09i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 + 3.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
19 \( 1 + (3.59 + 6.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.40 + 2.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 + (3.13 - 5.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.78 - 6.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.69T + 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 + (-4.20 - 7.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.52 + 2.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.655 + 1.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.96 + 8.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.89 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.160 - 0.277i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (-5.84 - 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.62T + 97T^{2} \)
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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

−9.288847244699246678665755498656, −8.766611191793969881886251233367, −8.131798867693394648516478006401, −6.63497608722207373043501851099, −6.23644514288711450517751040286, −4.85458393082434008720328233448, −3.86411887393945431317301225980, −2.99752688062502587664769472988, −1.28162084311339266286951332116, −0.54748981232875979672667132649, 2.05930851198371789818375675643, 3.36589432712910161009120410464, 3.95334050080774608684487985965, 5.83882789594136308519566387330, 6.23016726788751749066037047291, 7.06552552919803416414626506402, 7.86754259438038326017880801351, 8.591046702832684349941422570806, 9.321756206724644450722948048693, 10.28360871123324306727519924319

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