↑

Properties

Label 2-1170-13.10-c1-0-15
Degree $2$
Conductor $1170$
Sign $-0.252 + 0.967i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + i·5-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−2.59 + 1.5i)11-s + (−1 − 3.46i)13-s + (−0.5 − 0.866i)16-s + (−3 − 1.73i)19-s + (0.866 + 0.499i)20-s + (1.5 − 2.59i)22-s + (−0.866 − 1.5i)23-s − 25-s + (2.59 + 2.49i)26-s + (−0.866 − 1.5i)29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (−0.783 + 0.452i)11-s + (−0.277 − 0.960i)13-s + (−0.125 − 0.216i)16-s + (−0.688 − 0.397i)19-s + (0.193 + 0.111i)20-s + (0.319 − 0.553i)22-s + (−0.180 − 0.312i)23-s − 0.200·25-s + (0.509 + 0.490i)26-s + (−0.160 − 0.278i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4252206609\)
\(L(\frac12)\) \(\approx\) \(0.4252206609\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (1 + 3.46i)T \)
good7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 3i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + (7.79 + 4.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3 + 1.73i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.680306846967107841369039291516, −8.582383893790719759693083275951, −7.88711198848140809849313622373, −7.21432725574976184958728427928, −6.29846824628144305493188505908, −5.44867083845582109644410344733, −4.45728789059022702286325449096, −3.03838800239601637790430338817, −2.06859880090404520935780201036, −0.21928326095332361961473682522, 1.46441954361548520109945058475, 2.60203675634136363534359677463, 3.79701017572963325831829949385, 4.81122654184525279681477889232, 5.83941196901512329257800384829, 6.86409590290084049904587064450, 7.71809722320199365393166068827, 8.568277961101023087185559390937, 9.092495692747169899130473424919, 10.06785180338100962731039617633

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