Properties

Label 2-1014-13.10-c1-0-11
Degree $2$
Conductor $1014$
Sign $-0.0515 - 0.998i$
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 + 2i·5-s + (−0.866 − 0.499i)6-s + (3.46 + 2i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)10-s + (3.46 − 2i)11-s + 0.999·12-s − 3.99·14-s + (−1.73 + i)15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)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 + 0.894i·5-s + (−0.353 − 0.204i)6-s + (1.30 + 0.755i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (1.04 − 0.603i)11-s + 0.288·12-s − 1.06·14-s + (−0.447 + 0.258i)15-s + (−0.125 − 0.216i)16-s + (0.242 − 0.420i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.0515 - 0.998i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.0515 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.674657195\)
\(L(\frac12)\) \(\approx\) \(1.674657195\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 5T^{2} \)
7 \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.46 + 2i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.92 - 4i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.66 - 5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-3.46 - 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.8 - 8i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.92 - 4i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + (12.1 - 7i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.66 + 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.936193974606221197965670422348, −9.387423561374392878630978863963, −8.436068708170155232079764512734, −7.87459788872227501240420878374, −6.95557565334194861427303394131, −5.87340093876526780398250693671, −5.18839788149181929584059259586, −3.86582321560939600417178721843, −2.76477429205027418481081924013, −1.49784687045497768096058282702, 1.14810231307952175467939165403, 1.60290371862982725135754429192, 3.27624232899731830590955182173, 4.45597645166152038457802064950, 5.19794544272011510995124710335, 6.72617146178320099045695754933, 7.45597155505084620545421606239, 8.105933640065323688362981131644, 8.988310900441996454595398524526, 9.464888196908609242086131943348

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