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Properties

Label 2-1191-1191.1148-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.888 + 0.459i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.888 + 0.458i)3-s + (−0.654 + 0.755i)4-s + (−0.580 − 0.814i)7-s + (0.580 − 0.814i)9-s + (0.235 − 0.971i)12-s + (−0.473 − 1.36i)13-s + (−0.142 − 0.989i)16-s + (1.56 + 1.23i)19-s + (0.888 + 0.458i)21-s + (−0.327 − 0.945i)25-s + (−0.142 + 0.989i)27-s + (0.995 + 0.0950i)28-s + (1.21 − 0.782i)31-s + (0.235 + 0.971i)36-s + (−0.0135 − 0.284i)37-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)3-s + (−0.654 + 0.755i)4-s + (−0.580 − 0.814i)7-s + (0.580 − 0.814i)9-s + (0.235 − 0.971i)12-s + (−0.473 − 1.36i)13-s + (−0.142 − 0.989i)16-s + (1.56 + 1.23i)19-s + (0.888 + 0.458i)21-s + (−0.327 − 0.945i)25-s + (−0.142 + 0.989i)27-s + (0.995 + 0.0950i)28-s + (1.21 − 0.782i)31-s + (0.235 + 0.971i)36-s + (−0.0135 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.888 + 0.459i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (1148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.888 + 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5638962869\)
\(L(\frac12)\) \(\approx\) \(0.5638962869\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.888 - 0.458i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.654 - 0.755i)T^{2} \)
5 \( 1 + (0.327 + 0.945i)T^{2} \)
7 \( 1 + (0.580 + 0.814i)T + (-0.327 + 0.945i)T^{2} \)
11 \( 1 + (0.888 + 0.458i)T^{2} \)
13 \( 1 + (0.473 + 1.36i)T + (-0.786 + 0.618i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-1.56 - 1.23i)T + (0.235 + 0.971i)T^{2} \)
23 \( 1 + (0.888 + 0.458i)T^{2} \)
29 \( 1 + (-0.981 - 0.189i)T^{2} \)
31 \( 1 + (-1.21 + 0.782i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.0135 + 0.284i)T + (-0.995 + 0.0950i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.928 - 0.371i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.580 - 0.814i)T^{2} \)
61 \( 1 + (-0.581 + 1.67i)T + (-0.786 - 0.618i)T^{2} \)
67 \( 1 + (0.0623 + 1.30i)T + (-0.995 + 0.0950i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.607 + 0.243i)T + (0.723 + 0.690i)T^{2} \)
79 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.654 - 0.755i)T^{2} \)
89 \( 1 + (0.786 + 0.618i)T^{2} \)
97 \( 1 + (1.84 - 0.176i)T + (0.981 - 0.189i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.934393013473642101243899892263, −9.409142806812162343669577429282, −7.997557027452576365615164785623, −7.62775380874118696480399775365, −6.48467690555499999345330060205, −5.57048934088878877034426576925, −4.70929985893367563420176464410, −3.80265266413079888493378123180, −3.07767867229045306920850571229, −0.66261793968660141775185459631, 1.25124623077327169092930831551, 2.60351346525455686081008542058, 4.22900601674622396765251860029, 5.15470455016201579422247852386, 5.66860007818123828809834742788, 6.67948487154325487392472474435, 7.21745027339993635334103346415, 8.596492557647412683851101357305, 9.385450201660117887828340858122, 9.844216791866574332557113043409

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