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Properties

Label 2-1014-13.10-c1-0-23
Degree $2$
Conductor $1014$
Sign $-0.982 - 0.188i$
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 − 4.04i·5-s + (−0.866 − 0.499i)6-s + (−0.599 − 0.346i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−2.02 − 3.50i)10-s + (−4.20 + 2.42i)11-s − 0.999·12-s − 0.692·14-s + (−3.50 + 2.02i)15-s + (−0.5 − 0.866i)16-s + (3.69 − 6.39i)17-s + 0.999i·18-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s − 1.81i·5-s + (−0.353 − 0.204i)6-s + (−0.226 − 0.130i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.640 − 1.10i)10-s + (−1.26 + 0.731i)11-s − 0.288·12-s − 0.184·14-s + (−0.905 + 0.522i)15-s + (−0.125 − 0.216i)16-s + (0.895 − 1.55i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.982 - 0.188i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428663208\)
\(L(\frac12)\) \(\approx\) \(1.428663208\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.04iT - 5T^{2} \)
7 \( 1 + (0.599 + 0.346i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.20 - 2.42i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.69 + 6.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 + 0.890i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.55 - 4.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.67 - 2.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.972iT - 31T^{2} \)
37 \( 1 + (-1.11 + 0.643i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.30 - 0.753i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.15 - 7.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.20iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + (1.13 + 0.653i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.198 + 0.343i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.24 - 3.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.15 + 0.664i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.65iT - 73T^{2} \)
79 \( 1 + 8.33T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + (-2.69 + 1.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.39 + 4.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.631260364274509785821232824309, −8.721310479164367845664767041362, −7.75052637126609831209686763919, −7.05479718587607731149683332711, −5.62308695619835490531035964524, −5.14160715222956178084571281496, −4.50996480180086765671269642910, −3.03912022321123794418160505883, −1.73590671613065398927896636591, −0.53244452383544422801633605521, 2.51169858205239907591643534146, 3.23413934408203532929953114074, 4.08898470965568965029908795563, 5.48108964738374546641391415390, 6.07870243032951567253933578998, 6.78627411926359045019214918633, 7.78337409643222521903228571307, 8.489698729819357599574461250919, 9.974162460866962006130862153378, 10.59465104421944326567177486370

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