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Properties

Label 2-1014-13.4-c1-0-19
Degree $2$
Conductor $1014$
Sign $-0.265 + 0.964i$
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 − 3.73i·5-s + (−0.866 + 0.499i)6-s + (−2.36 + 1.36i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.86 − 3.23i)10-s + (−1.09 − 0.633i)11-s − 0.999·12-s − 2.73·14-s + (3.23 + 1.86i)15-s + (−0.5 + 0.866i)16-s + (−2.86 − 4.96i)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 − 1.66i·5-s + (−0.353 + 0.204i)6-s + (−0.894 + 0.516i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.590 − 1.02i)10-s + (−0.331 − 0.191i)11-s − 0.288·12-s − 0.730·14-s + (0.834 + 0.481i)15-s + (−0.125 + 0.216i)16-s + (−0.695 − 1.20i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.265 + 0.964i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.265 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8551107632\)
\(L(\frac12)\) \(\approx\) \(0.8551107632\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.73iT - 5T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.09 + 0.633i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.86 + 4.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.09 + 3.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 + 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (3.06 + 1.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.13 + 4.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.83 + 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.09 - 2.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.26iT - 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 - 0.196iT - 83T^{2} \)
89 \( 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.566142673412678641061300229868, −8.791966465346618534900672602801, −8.329884144295394286809012506481, −6.95342061251411452913824849256, −6.05946124444720685329282293368, −5.23399235460784412676336116036, −4.65214218448244023985792314115, −3.68702409204954292201059605349, −2.34213998501703105263335673941, −0.30343879879023789325653856103, 1.90140886989601695175632634054, 3.00471344361325431322676938020, 3.67662596668744703938691514945, 4.96930059717601724539577696471, 6.39283012958816740439915082328, 6.52008279617424131326157016970, 7.27905551182654011068928738535, 8.423333808405596985681771643899, 9.826244437158319361769321732547, 10.40972739524467484745132970690

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