Properties

Label 2-1170-13.12-c1-0-23
Degree $2$
Conductor $1170$
Sign $-1$
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  i·2-s − 4-s i·5-s − 4.60i·7-s + i·8-s − 10-s + 3.60·13-s − 4.60·14-s + 16-s − 4.60·17-s − 4.60i·19-s + i·20-s + 1.39·23-s − 25-s − 3.60i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 1.74i·7-s + 0.353i·8-s − 0.316·10-s + 1.00·13-s − 1.23·14-s + 0.250·16-s − 1.11·17-s − 1.05i·19-s + 0.223i·20-s + 0.290·23-s − 0.200·25-s − 0.707i·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.170007064\)
\(L(\frac12)\) \(\approx\) \(1.170007064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 - 3.60T \)
good7 \( 1 + 4.60iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 + 4.60iT - 19T^{2} \)
23 \( 1 - 1.39T + 23T^{2} \)
29 \( 1 + 4.60T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + 9.21iT - 37T^{2} \)
41 \( 1 + 3.21iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 9.21iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 9.21iT - 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 3.21iT - 67T^{2} \)
71 \( 1 + 9.21iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 2.78iT - 83T^{2} \)
89 \( 1 + 15.2iT - 89T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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

−9.252645784952432429332199795045, −8.891767460786387990176091042533, −7.69396526152934103147073935464, −7.03274628397051416972650465303, −5.95308081148676960619149608294, −4.63297653432515527542396250964, −4.16042112515793033810378920628, −3.14775777238510614307352411497, −1.62545250391783214636567988570, −0.51487114833764607901905081384, 1.90099776105065766427192433005, 3.06447311149039710998548026074, 4.21200496971249717896478680118, 5.40331199792332387568937544010, 6.04304329254448087229724277265, 6.64149625038635540495788379015, 7.85416547768879583175197780591, 8.525544310527710369167190329911, 9.161138011573199076177604006441, 9.945491203669175515547786614445

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