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Properties

Label 2-546-273.185-c1-0-37
Degree $2$
Conductor $546$
Sign $-0.980 - 0.198i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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 + (1.11 − 1.32i)3-s + (0.499 + 0.866i)4-s + (−2.12 − 3.68i)5-s + (−1.62 + 0.592i)6-s + (−1.07 − 2.41i)7-s − 0.999i·8-s + (−0.519 − 2.95i)9-s + 4.25i·10-s − 1.48i·11-s + (1.70 + 0.301i)12-s + (2.41 + 2.67i)13-s + (−0.281 + 2.63i)14-s + (−7.26 − 1.28i)15-s + (−0.5 + 0.866i)16-s + (1.33 + 2.30i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.642 − 0.765i)3-s + (0.249 + 0.433i)4-s + (−0.952 − 1.64i)5-s + (−0.664 + 0.241i)6-s + (−0.405 − 0.914i)7-s − 0.353i·8-s + (−0.173 − 0.984i)9-s + 1.34i·10-s − 0.448i·11-s + (0.492 + 0.0869i)12-s + (0.669 + 0.742i)13-s + (−0.0752 + 0.703i)14-s + (−1.87 − 0.331i)15-s + (−0.125 + 0.216i)16-s + (0.323 + 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.980 - 0.198i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0893165 + 0.890151i\)
\(L(\frac12)\) \(\approx\) \(0.0893165 + 0.890151i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (1.07 + 2.41i)T \)
13 \( 1 + (-2.41 - 2.67i)T \)
good5 \( 1 + (2.12 + 3.68i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.48iT - 11T^{2} \)
17 \( 1 + (-1.33 - 2.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.10iT - 19T^{2} \)
23 \( 1 + (0.137 + 0.0792i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.413 - 0.238i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.04 - 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.23 + 9.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.13 + 8.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.00 + 6.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.27 + 3.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.23 - 1.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.39 - 4.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.64iT - 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + (-7.70 - 4.44i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.03 + 3.48i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.20 + 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.37T + 83T^{2} \)
89 \( 1 + (5.94 - 10.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.13 - 1.81i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.21125830121288000497353271765, −9.075514799636721529107719828179, −8.625325619990796922696673430198, −7.86187423509005814990901001814, −7.13634608923579111790492564060, −5.83949187650530121208519001994, −4.09302214815728621602633026339, −3.62178430995489133748316799912, −1.62436220348574942141742192007, −0.60612099423601440495779044953, 2.68985999986403211069457126171, 3.16564933986844178840029492914, 4.59328160856407390336213977589, 6.00637139362098245162121620303, 6.92586380789541119957806266414, 7.87889210294400569073294145609, 8.470073364945623273926519869323, 9.694348738797026886530217018429, 10.11387705973427292896535973028, 11.24499252394535341263094045625

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