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Properties

Label 2-1191-1191.941-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.440 - 0.897i$
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  + (0.981 + 0.189i)3-s + (−0.959 + 0.281i)4-s + (−0.928 + 0.371i)7-s + (0.928 + 0.371i)9-s + (−0.995 + 0.0950i)12-s + (0.839 + 0.800i)13-s + (0.841 − 0.540i)16-s + (−0.0748 + 1.57i)19-s + (−0.981 + 0.189i)21-s + (0.723 + 0.690i)25-s + (0.841 + 0.540i)27-s + (0.786 − 0.618i)28-s + (−0.759 − 0.876i)31-s + (−0.995 − 0.0950i)36-s + (−0.550 + 1.58i)37-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)3-s + (−0.959 + 0.281i)4-s + (−0.928 + 0.371i)7-s + (0.928 + 0.371i)9-s + (−0.995 + 0.0950i)12-s + (0.839 + 0.800i)13-s + (0.841 − 0.540i)16-s + (−0.0748 + 1.57i)19-s + (−0.981 + 0.189i)21-s + (0.723 + 0.690i)25-s + (0.841 + 0.540i)27-s + (0.786 − 0.618i)28-s + (−0.759 − 0.876i)31-s + (−0.995 − 0.0950i)36-s + (−0.550 + 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.440 - 0.897i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (941, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.440 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099447756\)
\(L(\frac12)\) \(\approx\) \(1.099447756\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 - 0.189i)T \)
397 \( 1 - T \)
good2 \( 1 + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.723 - 0.690i)T^{2} \)
7 \( 1 + (0.928 - 0.371i)T + (0.723 - 0.690i)T^{2} \)
11 \( 1 + (-0.981 + 0.189i)T^{2} \)
13 \( 1 + (-0.839 - 0.800i)T + (0.0475 + 0.998i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.0748 - 1.57i)T + (-0.995 - 0.0950i)T^{2} \)
23 \( 1 + (-0.981 + 0.189i)T^{2} \)
29 \( 1 + (-0.235 + 0.971i)T^{2} \)
31 \( 1 + (0.759 + 0.876i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.550 - 1.58i)T + (-0.786 - 0.618i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.888 + 0.458i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.928 + 0.371i)T^{2} \)
61 \( 1 + (-1.42 + 1.35i)T + (0.0475 - 0.998i)T^{2} \)
67 \( 1 + (-0.627 + 1.81i)T + (-0.786 - 0.618i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (1.28 + 0.663i)T + (0.580 + 0.814i)T^{2} \)
79 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.0475 + 0.998i)T^{2} \)
97 \( 1 + (-1.39 - 1.09i)T + (0.235 + 0.971i)T^{2} \)
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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

−9.825834087470158433816255201707, −9.197279400733966235569085535445, −8.630267806878410013098509947677, −7.918937912595408372634463016809, −6.89841746682930144554258317422, −5.88615050203511453985703302809, −4.77323809986333642496400285443, −3.68292127784582383068669416819, −3.34865506933185401116986422804, −1.78762032632115573009424459104, 0.981567419197658622985275271667, 2.74888214289349434704768542863, 3.59436745825879175082357142880, 4.43264607727260766773402969770, 5.54620137651988311349838775782, 6.63744065740164120188805132858, 7.36491447026359833819184568030, 8.583006167361375904707909575265, 8.783902213822761260943019417683, 9.748396501302139057303401196063

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