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Properties

Label 2-546-91.4-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.394 - 0.918i$
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  − i·2-s + (0.5 + 0.866i)3-s − 4-s + (−2.34 + 1.35i)5-s + (0.866 − 0.5i)6-s + (1.65 − 2.06i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (1.35 + 2.34i)10-s + (−4.41 + 2.54i)11-s + (−0.5 − 0.866i)12-s + (0.313 + 3.59i)13-s + (−2.06 − 1.65i)14-s + (−2.34 − 1.35i)15-s + 16-s − 5.73·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 + 0.499i)3-s − 0.5·4-s + (−1.04 + 0.604i)5-s + (0.353 − 0.204i)6-s + (0.627 − 0.778i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.427 + 0.740i)10-s + (−1.33 + 0.768i)11-s + (−0.144 − 0.249i)12-s + (0.0868 + 0.996i)13-s + (−0.550 − 0.443i)14-s + (−0.604 − 0.348i)15-s + 0.250·16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312512 + 0.474236i\)
\(L(\frac12)\) \(\approx\) \(0.312512 + 0.474236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.65 + 2.06i)T \)
13 \( 1 + (-0.313 - 3.59i)T \)
good5 \( 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.41 - 2.54i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (3.79 + 2.19i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.46T + 23T^{2} \)
29 \( 1 + (4.18 - 7.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.21 + 3.58i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.77iT - 37T^{2} \)
41 \( 1 + (-3.29 - 1.90i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.10 - 5.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.44 - 4.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.60 + 6.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.47iT - 59T^{2} \)
61 \( 1 + (-2.32 + 4.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.25 + 3.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.792 + 0.457i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.93 - 3.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.81 + 6.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + (8.20 - 4.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.11901969500587507552390545662, −10.54478201580486561984497905373, −9.394818990522004614373981133406, −8.532960127229662277164971800252, −7.53687496637132490153333630788, −6.88530512428058742332300920175, −4.86404455728406449040973260008, −4.41359974428833981841423645526, −3.32099198791523849061676769852, −2.09663465265701728166494637766, 0.30299746939185432956876707004, 2.44547401757216040483785278640, 3.88702057298882541158470276989, 5.10468523048639139143766031270, 5.78533580675331451730283960416, 7.15071329855684693748030004351, 8.035450817725814357834221648622, 8.417135080818796661815446223318, 9.117127711357585636798977570397, 10.76828444546844033084747078363

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