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Properties

Label 2-1190-119.16-c1-0-22
Degree $2$
Conductor $1190$
Sign $0.981 - 0.193i$
Analytic cond. $9.50219$
Root an. cond. $3.08256$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.825 − 0.476i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s − 0.953i·6-s + (0.410 + 2.61i)7-s − 0.999·8-s + (−1.04 + 1.81i)9-s + (0.866 − 0.499i)10-s + (−5.02 + 2.90i)11-s + (−0.825 − 0.476i)12-s + 2.77·13-s + (2.46 + 0.951i)14-s + 0.953·15-s + (−0.5 + 0.866i)16-s + (3.61 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.476 − 0.275i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s − 0.389i·6-s + (0.155 + 0.987i)7-s − 0.353·8-s + (−0.348 + 0.603i)9-s + (0.273 − 0.158i)10-s + (−1.51 + 0.874i)11-s + (−0.238 − 0.137i)12-s + 0.770·13-s + (0.659 + 0.254i)14-s + 0.246·15-s + (−0.125 + 0.216i)16-s + (0.877 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.981 - 0.193i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ 0.981 - 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.207473222\)
\(L(\frac12)\) \(\approx\) \(2.207473222\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.410 - 2.61i)T \)
17 \( 1 + (-3.61 + 1.97i)T \)
good3 \( 1 + (-0.825 + 0.476i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (5.02 - 2.90i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.77T + 13T^{2} \)
19 \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.52 - 2.61i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.68iT - 29T^{2} \)
31 \( 1 + (-5.32 + 3.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.37 - 4.25i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.456iT - 41T^{2} \)
43 \( 1 + 2.56T + 43T^{2} \)
47 \( 1 + (2.79 - 4.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.63 - 4.55i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.33 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.74 + 1.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.95 + 13.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (-8.48 + 4.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (13.3 + 7.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.13T + 83T^{2} \)
89 \( 1 + (0.791 - 1.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.20iT - 97T^{2} \)
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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

−9.767090147579735989008496083412, −9.117012497828150929798658401860, −8.129201784964704142858289397414, −7.53443536942325716728825987963, −6.26888761116957416342101896571, −5.24749279120836116925149752637, −4.89445147812557512059207438996, −3.03828065347359620682317667872, −2.70063416557524946537751217150, −1.58335688511273929114723925339, 0.837715044158322446735895057080, 2.81806224119558467500441429954, 3.58582092260783127668322753370, 4.55965324706609646473873344674, 5.64545346095663098392455385001, 6.18883182797378568315938535854, 7.35936063749937277066320981121, 8.220420059463677827679274999301, 8.568133369286496119923841574092, 9.750206498255918066009063990014

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