Properties

Label 2-546-273.152-c1-0-7
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: $\chi_{546} (425, \cdot )$
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

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

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