Properties

Label 2-1191-1191.821-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.939 - 0.342i$
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 + (−0.643 + 1.60i)7-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)12-s + (−1.12 − 1.17i)13-s + (0.841 + 0.540i)16-s + (−0.0748 − 1.57i)19-s + (−0.327 + 1.70i)21-s + (−0.723 + 0.690i)25-s + (0.841 − 0.540i)27-s + (−1.07 + 1.36i)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 + (−0.643 + 1.60i)7-s + (0.928 − 0.371i)9-s + (0.995 + 0.0950i)12-s + (−1.12 − 1.17i)13-s + (0.841 + 0.540i)16-s + (−0.0748 − 1.57i)19-s + (−0.327 + 1.70i)21-s + (−0.723 + 0.690i)25-s + (0.841 − 0.540i)27-s + (−1.07 + 1.36i)28-s + (−0.759 + 0.876i)31-s + (0.995 − 0.0950i)36-s + (0.550 + 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.636008028\)
\(L(\frac12)\) \(\approx\) \(1.636008028\)
\(L(1)\) 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 \( 1 + (0.643 - 1.60i)T + (-0.723 - 0.690i)T^{2} \)
11 \( 1 + (-0.981 - 0.189i)T^{2} \)
13 \( 1 + (1.12 + 1.17i)T + (-0.0475 + 0.998i)T^{2} \)
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 \( 1 + (0.261 - 0.273i)T + (-0.0475 - 0.998i)T^{2} \)
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

−9.818076905430799683500677442163, −9.111387418037217285724651858780, −8.404957597007536444968797799365, −7.51337459931896011795188709998, −6.88204449893027604520839499479, −5.90981098590562872291326665400, −4.98029799337640712767059727072, −3.29517158522341122944347324444, −2.80056082048587329863407250670, −2.02671623177416754082186617015, 1.60120302223673867464325490395, 2.61871643289529769231925013733, 3.81956104097195393153025089002, 4.33897895322789298454856985428, 5.91410406770594953832993457911, 6.85442939241744388360690429292, 7.46091944577196846615195080922, 7.966996120154299173177948724414, 9.460983455542511048564872064118, 9.835698636464433586152746343533

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