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Properties

Label 2-1191-1191.674-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.967 + 0.252i$
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 + (−1.41 − 1.00i)7-s + (0.580 + 0.814i)9-s + (−0.235 − 0.971i)12-s + (1.30 + 0.451i)13-s + (−0.142 + 0.989i)16-s + (1.56 − 1.23i)19-s + (0.793 + 1.53i)21-s + (0.327 − 0.945i)25-s + (−0.142 − 0.989i)27-s + (−0.164 − 1.72i)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 + (−1.41 − 1.00i)7-s + (0.580 + 0.814i)9-s + (−0.235 − 0.971i)12-s + (1.30 + 0.451i)13-s + (−0.142 + 0.989i)16-s + (1.56 − 1.23i)19-s + (0.793 + 1.53i)21-s + (0.327 − 0.945i)25-s + (−0.142 − 0.989i)27-s + (−0.164 − 1.72i)28-s + (1.21 + 0.782i)31-s + (−0.235 + 0.971i)36-s + (0.0135 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8688213180\)
\(L(\frac12)\) \(\approx\) \(0.8688213180\)
\(L(1)\) 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 \( 1 + (1.41 + 1.00i)T + (0.327 + 0.945i)T^{2} \)
11 \( 1 + (0.888 - 0.458i)T^{2} \)
13 \( 1 + (-1.30 - 0.451i)T + (0.786 + 0.618i)T^{2} \)
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 \( 1 + (-0.866 + 0.299i)T + (0.786 - 0.618i)T^{2} \)
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

−10.15407474632182873442153282004, −9.148147795119970823069247158940, −8.039774714484138287928981066223, −7.18710670432471001022733031936, −6.60614778420291175192121471004, −6.17078204019566758103832637413, −4.73729559620682780091752606181, −3.67756013802685419183013239246, −2.81294989315667182481420627925, −1.09228639971093273880120074185, 1.22126886167096375928948228949, 2.91401707519636749804567624987, 3.74637560353183719153864808457, 5.37827609524487928206918779861, 5.74397485808447399655399796180, 6.37411449547932918048691980167, 7.18208311614282070475493922440, 8.545006185115562293995653377378, 9.600775008423479123331491120490, 9.908301192222054687255669444751

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