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Properties

Label 2-546-21.5-c1-0-20
Degree $2$
Conductor $546$
Sign $0.848 - 0.529i$
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 + (−0.913 + 1.47i)3-s + (0.499 + 0.866i)4-s + (1.72 − 2.98i)5-s + (−1.52 + 0.817i)6-s + (2.50 + 0.857i)7-s + 0.999i·8-s + (−1.33 − 2.68i)9-s + (2.98 − 1.72i)10-s + (2.62 − 1.51i)11-s + (−1.73 − 0.0552i)12-s − i·13-s + (1.73 + 1.99i)14-s + (2.81 + 5.26i)15-s + (−0.5 + 0.866i)16-s + (−2.09 − 3.62i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.527 + 0.849i)3-s + (0.249 + 0.433i)4-s + (0.770 − 1.33i)5-s + (−0.623 + 0.333i)6-s + (0.946 + 0.324i)7-s + 0.353i·8-s + (−0.443 − 0.896i)9-s + (0.943 − 0.544i)10-s + (0.792 − 0.457i)11-s + (−0.499 − 0.0159i)12-s − 0.277i·13-s + (0.464 + 0.532i)14-s + (0.727 + 1.35i)15-s + (−0.125 + 0.216i)16-s + (−0.507 − 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07727 + 0.595643i\)
\(L(\frac12)\) \(\approx\) \(2.07727 + 0.595643i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.913 - 1.47i)T \)
7 \( 1 + (-2.50 - 0.857i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.62 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.09 + 3.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.40 - 0.812i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.25 - 2.45i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.66iT - 29T^{2} \)
31 \( 1 + (5.11 - 2.95i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.43 - 7.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.863T + 41T^{2} \)
43 \( 1 + 8.77T + 43T^{2} \)
47 \( 1 + (-1.21 + 2.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.28 + 1.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.71 + 4.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.02 - 1.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.37 + 7.58i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (6.83 - 3.94i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.13 + 8.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.53T + 83T^{2} \)
89 \( 1 + (-7.14 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−11.12457624222524915729013856730, −9.915485778533021403123641296210, −8.889315809198655309978695403986, −8.642724754060996901051985356499, −7.04376721429062800973844486924, −5.85302010992878290319866280606, −5.10598832577493253052770273110, −4.73890008200444831020142336897, −3.35094550421334482881104745288, −1.41364852925769805080589490308, 1.62391831479959980267810026565, 2.44290103643338518007967621784, 4.00202120355069360683629734270, 5.23825752212849242064203247849, 6.24744898605610365252231653664, 6.85422922682081611196788504333, 7.64798618178566421009798619039, 9.058736476694793307757550946847, 10.32219718497843236652938421160, 10.90791418856744412093234606673

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