↑

Properties

Label 2-1183-91.55-c0-0-2
Degree $2$
Conductor $1183$
Sign $0.309 + 0.950i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (−0.900 − 1.56i)2-s + (−1.12 + 1.94i)4-s + (0.5 − 0.866i)7-s + 2.24·8-s + (−0.5 + 0.866i)9-s + (0.623 + 1.07i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80·18-s + (1.12 − 1.94i)22-s + (0.222 + 0.385i)23-s + 25-s + (1.12 + 1.94i)28-s + (0.900 + 1.56i)29-s + (−0.500 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.900 − 1.56i)2-s + (−1.12 + 1.94i)4-s + (0.5 − 0.866i)7-s + 2.24·8-s + (−0.5 + 0.866i)9-s + (0.623 + 1.07i)11-s − 1.80·14-s + (−0.900 − 1.56i)16-s + 1.80·18-s + (1.12 − 1.94i)22-s + (0.222 + 0.385i)23-s + 25-s + (1.12 + 1.94i)28-s + (0.900 + 1.56i)29-s + (−0.500 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6584295721\)
\(L(\frac12)\) \(\approx\) \(0.6584295721\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.445T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−10.00793244046243213649156391407, −9.111059584941339415839102366499, −8.491406114338974116714043058695, −7.58826207743894780567849939451, −6.91742569854597909138340424204, −5.11659080082994981239921886668, −4.34021780926030482957117138265, −3.33788584364139720569565803846, −2.20779607367151173291302926636, −1.25964275518920428264942236711, 0.991618335817046637737027278586, 2.91253615867833475559991204488, 4.45798032177267507148707569997, 5.48946359573041397361250088043, 6.21999625896606380232087145696, 6.62017355059676412688363855014, 7.913877997525838482007925073446, 8.426294449300649921105468810155, 9.065484349248129786524372073931, 9.578897796886129832026143037833

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