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Properties

Label 2-546-273.146-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.990 - 0.140i$
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 + (1.22 − 1.21i)3-s + (0.499 − 0.866i)4-s − 1.88·5-s + (−0.454 + 1.67i)6-s + (−1.27 + 2.31i)7-s + 0.999i·8-s + (0.0240 − 2.99i)9-s + (1.62 − 0.940i)10-s + (−3.16 + 1.82i)11-s + (−0.441 − 1.67i)12-s + (−3.37 + 1.27i)13-s + (−0.0521 − 2.64i)14-s + (−2.31 + 2.29i)15-s + (−0.5 − 0.866i)16-s + (2.66 − 4.61i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.709 − 0.704i)3-s + (0.249 − 0.433i)4-s − 0.840·5-s + (−0.185 + 0.682i)6-s + (−0.482 + 0.875i)7-s + 0.353i·8-s + (0.00801 − 0.999i)9-s + (0.514 − 0.297i)10-s + (−0.954 + 0.551i)11-s + (−0.127 − 0.483i)12-s + (−0.935 + 0.352i)13-s + (−0.0139 − 0.706i)14-s + (−0.596 + 0.592i)15-s + (−0.125 − 0.216i)16-s + (0.646 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.990 - 0.140i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.990 - 0.140i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00257098 + 0.0363716i\)
\(L(\frac12)\) \(\approx\) \(0.00257098 + 0.0363716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 + 1.21i)T \)
7 \( 1 + (1.27 - 2.31i)T \)
13 \( 1 + (3.37 - 1.27i)T \)
good5 \( 1 + 1.88T + 5T^{2} \)
11 \( 1 + (3.16 - 1.82i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.66 + 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.90 + 2.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.85 - 1.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.94iT - 31T^{2} \)
37 \( 1 + (0.930 + 1.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.84 - 4.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.18 - 8.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.28T + 47T^{2} \)
53 \( 1 - 4.50iT - 53T^{2} \)
59 \( 1 + (-2.45 + 4.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.25 - 1.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 + 4.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.1 + 5.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.08iT - 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 + (4.82 + 8.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.4 + 6.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34330021123031932672978448699, −9.944222827930604521090543581887, −9.361828442990049577570602943187, −8.439930246962042817452879791964, −7.61040014982560104663250356602, −7.13618175371877575915593516001, −5.97232666805241125376518190117, −4.71925298287599258098086177414, −3.09281794125188164030792140451, −2.14301739218652609120686471579, 0.02155916911286049997152168324, 2.34888084047384251056069515975, 3.60408011088407574714494981163, 4.14657078257793918859497457456, 5.66459092585510823040734937729, 7.23847991698211402388750732567, 7.980947604690264688247261291972, 8.440343544288220152101031869511, 9.729374588262045977546903627876, 10.38277635126957566503886487099

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