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Properties

Label 2-1191-1191.965-c0-0-0
Degree $2$
Conductor $1191$
Sign $0.981 - 0.192i$
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  + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.723 + 0.690i)7-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)12-s + (−0.0311 − 0.653i)13-s + (0.415 − 0.909i)16-s + (−0.469 − 0.0448i)19-s + (−0.928 + 0.371i)21-s + (0.0475 + 0.998i)25-s + (0.415 + 0.909i)27-s + (−0.235 + 0.971i)28-s + (0.0930 − 0.647i)31-s + (0.981 + 0.189i)36-s + (−0.653 − 0.513i)37-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)3-s + (0.841 − 0.540i)4-s + (−0.723 + 0.690i)7-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)12-s + (−0.0311 − 0.653i)13-s + (0.415 − 0.909i)16-s + (−0.469 − 0.0448i)19-s + (−0.928 + 0.371i)21-s + (0.0475 + 0.998i)25-s + (0.415 + 0.909i)27-s + (−0.235 + 0.971i)28-s + (0.0930 − 0.647i)31-s + (0.981 + 0.189i)36-s + (−0.653 − 0.513i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1191\)    =    \(3 \cdot 397\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(0.594386\)
Root analytic conductor: \(0.770964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1191} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1191,\ (\ :0),\ 0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593955947\)
\(L(\frac12)\) \(\approx\) \(1.593955947\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 - 0.371i)T \)
397 \( 1 - T \)
good2 \( 1 + (-0.841 + 0.540i)T^{2} \)
5 \( 1 + (-0.0475 - 0.998i)T^{2} \)
7 \( 1 + (0.723 - 0.690i)T + (0.0475 - 0.998i)T^{2} \)
11 \( 1 + (-0.928 + 0.371i)T^{2} \)
13 \( 1 + (0.0311 + 0.653i)T + (-0.995 + 0.0950i)T^{2} \)
17 \( 1 + (-0.841 - 0.540i)T^{2} \)
19 \( 1 + (0.469 + 0.0448i)T + (0.981 + 0.189i)T^{2} \)
23 \( 1 + (-0.928 + 0.371i)T^{2} \)
29 \( 1 + (0.888 + 0.458i)T^{2} \)
31 \( 1 + (-0.0930 + 0.647i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.653 + 0.513i)T + (0.235 + 0.971i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.38 - 0.407i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.580 - 0.814i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.723 + 0.690i)T^{2} \)
61 \( 1 + (-0.0883 + 1.85i)T + (-0.995 - 0.0950i)T^{2} \)
67 \( 1 + (1.32 + 1.04i)T + (0.235 + 0.971i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.0552 - 0.0775i)T + (-0.327 + 0.945i)T^{2} \)
79 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.995 + 0.0950i)T^{2} \)
97 \( 1 + (-0.273 - 1.12i)T + (-0.888 + 0.458i)T^{2} \)
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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

−9.856704922786679640010936961913, −9.327290475386901350631953010341, −8.401562191726641924798966467134, −7.55248854325692112535856238648, −6.69254036384868958751351330742, −5.80554818979944405848507553347, −4.92873072804877959191987324596, −3.52004709051923403782470444367, −2.79076149193779842996471896876, −1.80376041372929636577519174896, 1.65857628280749287642602298000, 2.75564678009579722560196503783, 3.56649780047270782934041793389, 4.42081590316618620931900649823, 6.15402254233097211142754481562, 6.86294065773856586112043248785, 7.30701498689739814743881613138, 8.331032463648161069151111200790, 8.877466747360588248193394384673, 10.06241852403456090138632801539

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