↑

Properties

Label 2-1191-1191.167-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.948 + 0.318i$
Analytic cond. $0.594386$
Root an. cond. $0.770964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (0.830 + 1.81i)7-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)12-s + (1.25 + 1.45i)13-s + (−0.142 + 0.989i)16-s + (−0.118 − 0.822i)19-s + (1.68 + 1.08i)21-s + (−0.654 − 0.755i)25-s + (−0.142 − 0.989i)27-s + (0.830 − 1.81i)28-s + (−1.61 − 1.03i)31-s + (−0.959 + 0.281i)36-s + (−0.239 − 0.153i)37-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (0.830 + 1.81i)7-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)12-s + (1.25 + 1.45i)13-s + (−0.142 + 0.989i)16-s + (−0.118 − 0.822i)19-s + (1.68 + 1.08i)21-s + (−0.654 − 0.755i)25-s + (−0.142 − 0.989i)27-s + (0.830 − 1.81i)28-s + (−1.61 − 1.03i)31-s + (−0.959 + 0.281i)36-s + (−0.239 − 0.153i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.948 + 0.318i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.948 + 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340859270\)
\(L(\frac12)\) \(\approx\) \(1.340859270\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 + 0.540i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (-0.830 - 1.81i)T + (-0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.118 + 0.822i)T + (-0.959 + 0.281i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.142 - 0.989i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
79 \( 1 - 1.68T + T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
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

−9.359585732130122586024197191889, −9.052290475795483691408402489245, −8.616432064613879885545681768228, −7.66354529734463481400506361979, −6.35777121101624358027621036704, −5.86364936702481199652414285799, −4.76178156202197969192031340209, −3.82039195904164731945942527921, −2.28233733103318338606332021302, −1.66245814528749711571086417319, 1.44011137931846510528246284782, 3.37780995134436051342500886278, 3.67927860329369719648442915840, 4.56152411666013433370545866384, 5.49089322653225736679160291668, 7.12625800999690553020761818049, 7.88236249232360041786077355795, 8.143398318997170645751027468227, 9.076355675431322980180281532567, 10.03191399034658039911800155253

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