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Properties

Label 2-546-273.38-c1-0-9
Degree $2$
Conductor $546$
Sign $-0.887 - 0.461i$
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  + (0.5 + 0.866i)2-s + (−1.62 + 0.611i)3-s + (−0.499 + 0.866i)4-s + (−0.0759 + 0.0438i)5-s + (−1.33 − 1.09i)6-s + (2.62 + 0.361i)7-s − 0.999·8-s + (2.25 − 1.98i)9-s + (−0.0759 − 0.0438i)10-s + (−2.83 + 4.90i)11-s + (0.280 − 1.70i)12-s + (2.43 + 2.66i)13-s + (0.997 + 2.45i)14-s + (0.0962 − 0.117i)15-s + (−0.5 − 0.866i)16-s + (−1.33 + 2.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.935 + 0.352i)3-s + (−0.249 + 0.433i)4-s + (−0.0339 + 0.0196i)5-s + (−0.546 − 0.448i)6-s + (0.990 + 0.136i)7-s − 0.353·8-s + (0.750 − 0.660i)9-s + (−0.0240 − 0.0138i)10-s + (−0.853 + 1.47i)11-s + (0.0811 − 0.493i)12-s + (0.674 + 0.738i)13-s + (0.266 + 0.654i)14-s + (0.0248 − 0.0303i)15-s + (−0.125 − 0.216i)16-s + (−0.323 + 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.887 - 0.461i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255739 + 1.04589i\)
\(L(\frac12)\) \(\approx\) \(0.255739 + 1.04589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.62 - 0.611i)T \)
7 \( 1 + (-2.62 - 0.361i)T \)
13 \( 1 + (-2.43 - 2.66i)T \)
good5 \( 1 + (0.0759 - 0.0438i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.83 - 4.90i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.33 - 2.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.32 + 5.76i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 - 0.613i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.84iT - 29T^{2} \)
31 \( 1 + (-0.441 + 0.765i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.456 - 0.263i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.99iT - 41T^{2} \)
43 \( 1 + 9.69T + 43T^{2} \)
47 \( 1 + (7.45 - 4.30i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.1 - 7.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.11 - 1.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.01 + 1.16i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.54 - 3.77i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.65T + 71T^{2} \)
73 \( 1 + (-4.11 + 7.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.19 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.86iT - 83T^{2} \)
89 \( 1 + (-2.93 + 1.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−11.17334630172790054357692400891, −10.49948292263765347588548452656, −9.368716206143016660316402707213, −8.453471979987046424073721785705, −7.28218555653557672504558793494, −6.67650081362284630055067822943, −5.42942591103997314587936851232, −4.79286238986932650147889266841, −3.98662562762353780923290652077, −1.93513070138120813221412074692, 0.62843713446541085237468009620, 2.10651616592652920761740882171, 3.67534951387322118080480388089, 4.86735891366043545283382077640, 5.67914515981989813323353252383, 6.41930238738884336180856032289, 8.096987579876297347882240738508, 8.254753304947803665509782281384, 10.12543122525039082171904905725, 10.56421669231285338122159182395

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