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Properties

Label 2-1190-119.16-c1-0-17
Degree $2$
Conductor $1190$
Sign $0.895 - 0.444i$
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 + (−2.06 + 1.19i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 2.38i·6-s + (−2.05 − 1.66i)7-s − 0.999·8-s + (1.33 − 2.31i)9-s + (0.866 − 0.499i)10-s + (−1.02 + 0.589i)11-s + (2.06 + 1.19i)12-s + 4.84·13-s + (−2.47 + 0.946i)14-s − 2.38·15-s + (−0.5 + 0.866i)16-s + (−4.12 + 0.126i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−1.19 + 0.687i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.972i·6-s + (−0.776 − 0.629i)7-s − 0.353·8-s + (0.445 − 0.770i)9-s + (0.273 − 0.158i)10-s + (−0.307 + 0.177i)11-s + (0.595 + 0.343i)12-s + 1.34·13-s + (−0.660 + 0.253i)14-s − 0.614·15-s + (−0.125 + 0.216i)16-s + (−0.999 + 0.0307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.895 - 0.444i$
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: \((2,\ 1190,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045646678\)
\(L(\frac12)\) \(\approx\) \(1.045646678\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 1.66i)T \)
17 \( 1 + (4.12 - 0.126i)T \)
good3 \( 1 + (2.06 - 1.19i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.02 - 0.589i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
19 \( 1 + (0.0242 - 0.0419i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.240 + 0.138i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + (-8.74 + 5.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 - 2.16i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.20iT - 41T^{2} \)
43 \( 1 + 2.79T + 43T^{2} \)
47 \( 1 + (0.362 - 0.628i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.33 - 9.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.18 - 2.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.5 - 6.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 - 9.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.665iT - 71T^{2} \)
73 \( 1 + (1.69 - 0.976i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-11.1 - 6.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.15T + 83T^{2} \)
89 \( 1 + (-3.13 + 5.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−10.18364627070826314364692857548, −9.388588686460171364321499798602, −8.407404855795791482371604112453, −6.89672607926858001558857430873, −6.27759065462892583202015595737, −5.55788167958219960740377432034, −4.55229459561611287008089971745, −3.85820318694622516183218532572, −2.70572021394751320248411677688, −1.00687503219415609831876494912, 0.59535481969872059280373408405, 2.28410642204526919387900823975, 3.67821740940527005219700472188, 4.92747462377307114865715916097, 5.71567199093258092575359599297, 6.47611408937863623793741438619, 6.57782188932887584733773079836, 8.006172149178143639838431968015, 8.710974185762682425059576231639, 9.632497864028134273080465567298

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