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Properties

Label 2-1014-13.10-c1-0-1
Degree $2$
Conductor $1014$
Sign $-0.309 + 0.951i$
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 + 4.29i·5-s + (−0.866 − 0.499i)6-s + (−3.77 − 2.17i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−2.14 − 3.72i)10-s + (−1.00 + 0.579i)11-s + 0.999·12-s + 4.35·14-s + (−3.72 + 2.14i)15-s + (−0.5 − 0.866i)16-s + (0.246 − 0.427i)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 + 1.92i·5-s + (−0.353 − 0.204i)6-s + (−1.42 − 0.823i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.679 − 1.17i)10-s + (−0.302 + 0.174i)11-s + 0.288·12-s + 1.16·14-s + (−0.960 + 0.554i)15-s + (−0.125 − 0.216i)16-s + (0.0599 − 0.103i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.309 + 0.951i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.309 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1751551732\)
\(L(\frac12)\) \(\approx\) \(0.1751551732\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.29iT - 5T^{2} \)
7 \( 1 + (3.77 + 2.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.00 - 0.579i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.246 + 0.427i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 + 0.890i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + 2.22iT - 31T^{2} \)
37 \( 1 + (-3.35 + 1.93i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.47 - 3.15i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.69 + 6.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.78iT - 47T^{2} \)
53 \( 1 + 2.51T + 53T^{2} \)
59 \( 1 + (-5.74 - 3.31i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.24 - 9.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.54 + 2.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.12 + 4.69i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.374iT - 73T^{2} \)
79 \( 1 - 2.65T + 79T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (0.723 - 0.417i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.9 + 9.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.29370694170914245339767615851, −9.880812664317270238423263299489, −9.155646087720270451747229825823, −7.73544963796763095441969699859, −7.25678416644741351045671835089, −6.50266543253026623644561998092, −5.80841800206269669139266246548, −4.08020582291463336337554917394, −3.24951422992323384969600386048, −2.42184522892591391666355277332, 0.092556461876884014049368779289, 1.41764159237509913736777766700, 2.65952403619269291300121019298, 3.78595798668617716415283655014, 5.09202852438830089461063150188, 5.96461120132338015638387979140, 6.92134819215543182582475748848, 8.177290132048395271791952678295, 8.583952533073293458947480412922, 9.342505986612330217333779383448

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