Properties

Label 2-1170-13.10-c1-0-4
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 + (−1.73 − i)7-s − 0.999i·8-s + (0.5 + 0.866i)10-s + (−5.59 + 3.23i)11-s + (1 + 3.46i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (6.46 + 3.73i)19-s + (0.866 + 0.499i)20-s + (−3.23 + 5.59i)22-s + (1.86 + 3.23i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−0.654 − 0.377i)7-s − 0.353i·8-s + (0.158 + 0.273i)10-s + (−1.68 + 0.974i)11-s + (0.277 + 0.960i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (1.48 + 0.856i)19-s + (0.193 + 0.111i)20-s + (−0.689 + 1.19i)22-s + (0.389 + 0.673i)23-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

\(L(1)\) \(\approx\) \(1.479160334\)
\(L(\frac12)\) \(\approx\) \(1.479160334\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.59 - 3.23i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.133 - 0.232i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-7.96 + 4.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 - i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.96 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.53iT - 47T^{2} \)
53 \( 1 + 0.928T + 53T^{2} \)
59 \( 1 + (7.33 + 4.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.19 + 9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.92 - 5.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.7 - 6.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 8.92iT - 83T^{2} \)
89 \( 1 + (-0.464 + 0.267i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.464 - 0.267i)T + (48.5 + 84.0i)T^{2} \)
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

−9.891843289933341365350299688936, −9.621966532351997157303673096828, −8.129244151766915709580112428078, −7.34653466108674566726061097254, −6.59009554333405846030602547046, −5.63941808434867767800243329490, −4.73696865178550778279771079236, −3.72130937746582561100770843859, −2.84514183161256091429447935564, −1.69914196548426895204772195208, 0.49960030530360120169901087369, 2.78660213021323231316526761286, 3.09536516962179707959667223381, 4.67950811094152170966627093169, 5.38896975615607281424175572036, 5.98145080846933360432911491234, 7.12422473548658492877037738453, 7.915078872514810074591563557058, 8.651899310974210326949188784055, 9.526061882375803740841809372095

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