Properties

Label 2-1170-1.1-c1-0-16
Degree $2$
Conductor $1170$
Sign $-1$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 2·11-s − 13-s + 4·14-s + 16-s − 2·17-s + 6·19-s + 20-s − 2·22-s − 6·23-s + 25-s + 26-s − 4·28-s − 2·29-s − 6·31-s − 32-s + 2·34-s − 4·35-s − 2·37-s − 6·38-s − 40-s − 10·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.223·20-s − 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 0.371·29-s − 1.07·31-s − 0.176·32-s + 0.342·34-s − 0.676·35-s − 0.328·37-s − 0.973·38-s − 0.158·40-s − 1.56·41-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: yes
Arithmetic: yes
Character: $\chi_{1170} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−9.406053246876928691493545957016, −8.868774471216060607456403615280, −7.66670900603824463060194393110, −6.87025308528864829107350186696, −6.22340654796584066361422256816, −5.34711726354022861021902847980, −3.81800601968201289194081442487, −2.96623355538274298367536210534, −1.69050510807477611643559790122, 0, 1.69050510807477611643559790122, 2.96623355538274298367536210534, 3.81800601968201289194081442487, 5.34711726354022861021902847980, 6.22340654796584066361422256816, 6.87025308528864829107350186696, 7.66670900603824463060194393110, 8.868774471216060607456403615280, 9.406053246876928691493545957016

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