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Properties

Label 2-182-91.88-c1-0-5
Degree $2$
Conductor $182$
Sign $0.767 - 0.641i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (0.866 + 0.5i)2-s + 1.40·3-s + (0.499 + 0.866i)4-s + (−0.552 + 0.319i)5-s + (1.21 + 0.703i)6-s + (−0.103 + 2.64i)7-s + 0.999i·8-s − 1.02·9-s − 0.638·10-s − 4.54i·11-s + (0.703 + 1.21i)12-s + (2.48 − 2.61i)13-s + (−1.41 + 2.23i)14-s + (−0.777 + 0.448i)15-s + (−0.5 + 0.866i)16-s + (0.354 + 0.613i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + 0.811·3-s + (0.249 + 0.433i)4-s + (−0.247 + 0.142i)5-s + (0.497 + 0.287i)6-s + (−0.0392 + 0.999i)7-s + 0.353i·8-s − 0.340·9-s − 0.201·10-s − 1.37i·11-s + (0.202 + 0.351i)12-s + (0.688 − 0.725i)13-s + (−0.377 + 0.598i)14-s + (−0.200 + 0.115i)15-s + (−0.125 + 0.216i)16-s + (0.0859 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $0.767 - 0.641i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ 0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75158 + 0.635424i\)
\(L(\frac12)\) \(\approx\) \(1.75158 + 0.635424i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.103 - 2.64i)T \)
13 \( 1 + (-2.48 + 2.61i)T \)
good3 \( 1 - 1.40T + 3T^{2} \)
5 \( 1 + (0.552 - 0.319i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.54iT - 11T^{2} \)
17 \( 1 + (-0.354 - 0.613i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 + (3.25 - 5.63i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.63 + 6.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.75 - 3.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.82 - 1.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.62 + 2.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.21 - 9.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.68 - 0.972i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.20 + 5.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.87 - 1.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.83T + 61T^{2} \)
67 \( 1 - 3.92iT - 67T^{2} \)
71 \( 1 + (9.11 + 5.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.55 - 3.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.48 - 9.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.41iT - 83T^{2} \)
89 \( 1 + (-4.79 - 2.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.96 + 4.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.23682872816178579351699176474, −11.64450156623077904411326731501, −11.22665805404129057136570444227, −9.428726274511940012062638954958, −8.495824037909295770240165733029, −7.84870119764123390286962879612, −6.21625549513982940724490831700, −5.41129594371211237548509037444, −3.57774580050629020066226320773, −2.72220273343157236347758397105, 1.98716291403814584283504598023, 3.67372022877390617836175657218, 4.45645208273136498580664526144, 6.17872434835218983564944457365, 7.42405800851026301534885500877, 8.413195429791166736755174468521, 9.726140363430051821898923279963, 10.50109628814876066301871806915, 11.76350402516444545867765531920, 12.56516992367980721478576053625

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