Properties

Label 2-1911-1911.89-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.956 - 0.290i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
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  + (−0.930 + 0.365i)3-s + (0.433 − 0.900i)4-s + (−0.680 + 0.733i)7-s + (0.733 − 0.680i)9-s + (−0.0747 + 0.997i)12-s + (−0.563 + 0.826i)13-s + (−0.623 − 0.781i)16-s + (1.93 + 0.517i)19-s + (0.365 − 0.930i)21-s + (0.680 + 0.733i)25-s + (−0.433 + 0.900i)27-s + (0.365 + 0.930i)28-s + (0.638 + 0.170i)31-s + (−0.294 − 0.955i)36-s + (0.649 − 1.85i)37-s + ⋯
L(s)  = 1  + (−0.930 + 0.365i)3-s + (0.433 − 0.900i)4-s + (−0.680 + 0.733i)7-s + (0.733 − 0.680i)9-s + (−0.0747 + 0.997i)12-s + (−0.563 + 0.826i)13-s + (−0.623 − 0.781i)16-s + (1.93 + 0.517i)19-s + (0.365 − 0.930i)21-s + (0.680 + 0.733i)25-s + (−0.433 + 0.900i)27-s + (0.365 + 0.930i)28-s + (0.638 + 0.170i)31-s + (−0.294 − 0.955i)36-s + (0.649 − 1.85i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8801901222\)
\(L(\frac12)\) \(\approx\) \(0.8801901222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.930 - 0.365i)T \)
7 \( 1 + (0.680 - 0.733i)T \)
13 \( 1 + (0.563 - 0.826i)T \)
good2 \( 1 + (-0.433 + 0.900i)T^{2} \)
5 \( 1 + (-0.680 - 0.733i)T^{2} \)
11 \( 1 + (-0.997 + 0.0747i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.365 - 0.930i)T^{2} \)
31 \( 1 + (-0.638 - 0.170i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.649 + 1.85i)T + (-0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.294 + 0.955i)T^{2} \)
43 \( 1 + (-0.202 - 1.34i)T + (-0.955 + 0.294i)T^{2} \)
47 \( 1 + (0.997 - 0.0747i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.974 + 0.222i)T^{2} \)
61 \( 1 + (0.0841 - 0.123i)T + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (-1.91 + 0.514i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.930 - 0.365i)T^{2} \)
73 \( 1 + (-1.51 - 0.0566i)T + (0.997 + 0.0747i)T^{2} \)
79 \( 1 + (0.433 - 0.751i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.433 - 0.900i)T^{2} \)
89 \( 1 + (0.433 + 0.900i)T^{2} \)
97 \( 1 + (-0.0193 - 0.0722i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.577801794578329983965276524061, −9.128893515330980150563844692013, −7.59224159995348906010305412233, −6.83797208830701227408277972696, −6.17295302365266392259519540745, −5.42747908000238867799282289460, −4.91302791527706402838995594087, −3.64513757145296975614770185439, −2.47027120062093164635959278675, −1.13154392337542971087670130153, 0.904295930860383391477701547039, 2.59237165973171912684200446515, 3.40131978188612073315113047237, 4.52304235404837243920052080701, 5.35731429138748098518200059797, 6.43141493625630636420909448546, 6.99008082000943678558614476881, 7.60957970412730190175552045204, 8.279744583842741089600306240000, 9.585726672645685907095681889398

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