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Properties

Label 2-1014-13.10-c1-0-19
Degree $2$
Conductor $1014$
Sign $0.499 + 0.866i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.499 + 0.866i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.907070009\)
\(L(\frac12)\) \(\approx\) \(2.907070009\)
\(L(\frac{3}{2})\) 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

−9.574346410486782411616148080512, −9.040172399368037804491896001844, −8.357431226244508307344861618931, −7.62060142264917968103955846810, −5.77132799379380566911483272166, −5.29271792297439928224600933508, −4.61345036301370748362578998310, −3.82158863781274278146044499303, −2.19360292997510324133727918395, −1.23123882533038882980903292049, 1.74002214049556484415371017664, 2.86188016913223535896736737967, 3.77391869083153698542313634539, 4.79961459138898104700116236615, 6.07187280442768088775355593771, 6.81090519095553733445283688163, 7.55588977516304615720602147580, 7.84393969889433673616940392636, 9.193372352070152164568860589206, 10.44277524936217489465487958093

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