Properties

Label 2-1170-1.1-c1-0-7
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 yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s − 2.60·7-s + 8-s + 10-s + 13-s − 2.60·14-s + 16-s + 4.60·17-s + 6.60·19-s + 20-s + 4.60·23-s + 25-s + 26-s − 2.60·28-s − 4.60·29-s + 2·31-s + 32-s + 4.60·34-s − 2.60·35-s + 11.2·37-s + 6.60·38-s + 40-s + 3.21·41-s + 5.21·43-s + 4.60·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.984·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s − 0.696·14-s + 0.250·16-s + 1.11·17-s + 1.51·19-s + 0.223·20-s + 0.960·23-s + 0.200·25-s + 0.196·26-s − 0.492·28-s − 0.855·29-s + 0.359·31-s + 0.176·32-s + 0.789·34-s − 0.440·35-s + 1.84·37-s + 1.07·38-s + 0.158·40-s + 0.501·41-s + 0.794·43-s + 0.679·46-s + ⋯

Functional equation

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

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: $\chi_{1170} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.686641799\)
\(L(\frac12)\) \(\approx\) \(2.686641799\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 - 6.60T + 19T^{2} \)
23 \( 1 - 4.60T + 23T^{2} \)
29 \( 1 + 4.60T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 - 5.21T + 43T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 7.21T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 6.60T + 73T^{2} \)
79 \( 1 + 1.21T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 3.21T + 89T^{2} \)
97 \( 1 - 6.60T + 97T^{2} \)
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

−9.635986929923984894284432054030, −9.326358798062685494621007363038, −7.897317131235509701685615963060, −7.20750154291776503825351295184, −6.18339040129388593714578191135, −5.66479110705734512417140733146, −4.65590499977135051485604735436, −3.40746312614170244869524286466, −2.84591525267786939907063306852, −1.22386453611003018721553684630, 1.22386453611003018721553684630, 2.84591525267786939907063306852, 3.40746312614170244869524286466, 4.65590499977135051485604735436, 5.66479110705734512417140733146, 6.18339040129388593714578191135, 7.20750154291776503825351295184, 7.897317131235509701685615963060, 9.326358798062685494621007363038, 9.635986929923984894284432054030

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