Properties

Label 2-546-273.251-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.473 + 0.880i$
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 + (−1.72 + 0.182i)3-s + (−0.499 − 0.866i)4-s − 3.28i·5-s + (0.702 − 1.58i)6-s + (−0.203 + 2.63i)7-s + 0.999·8-s + (2.93 − 0.629i)9-s + (2.84 + 1.64i)10-s + (−2.08 + 3.61i)11-s + (1.01 + 1.40i)12-s + (3.60 + 0.0395i)13-s + (−2.18 − 1.49i)14-s + (0.599 + 5.65i)15-s + (−0.5 + 0.866i)16-s + (−3.84 − 6.66i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.994 + 0.105i)3-s + (−0.249 − 0.433i)4-s − 1.46i·5-s + (0.286 − 0.646i)6-s + (−0.0770 + 0.997i)7-s + 0.353·8-s + (0.977 − 0.209i)9-s + (0.899 + 0.519i)10-s + (−0.628 + 1.08i)11-s + (0.294 + 0.404i)12-s + (0.999 + 0.0109i)13-s + (−0.583 − 0.399i)14-s + (0.154 + 1.45i)15-s + (−0.125 + 0.216i)16-s + (−0.933 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.473 + 0.880i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.473 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148732 - 0.248790i\)
\(L(\frac12)\) \(\approx\) \(0.148732 - 0.248790i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.182i)T \)
7 \( 1 + (0.203 - 2.63i)T \)
13 \( 1 + (-3.60 - 0.0395i)T \)
good5 \( 1 + 3.28iT - 5T^{2} \)
11 \( 1 + (2.08 - 3.61i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.84 + 6.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.26 + 2.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.82 + 2.20i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.55 + 2.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 + (-5.29 - 3.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 + 1.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.64 + 8.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.60iT - 47T^{2} \)
53 \( 1 - 4.48iT - 53T^{2} \)
59 \( 1 + (1.69 - 0.978i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.38 - 3.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.107 + 0.0620i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.79 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.12T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 + 5.80iT - 83T^{2} \)
89 \( 1 + (-5.51 - 3.18i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.26 + 5.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.40061613091715245950497863915, −9.295199974383128753701811524397, −8.990850879768418389505981752324, −7.83564590484322002253858767617, −6.78633854702613615855731440525, −5.74617451372073465508924033939, −5.05138333893235557385523479087, −4.38285989703085196895828126790, −1.92749863530216958012376111435, −0.21138930254821718079906936664, 1.68728773943785321007340405421, 3.39736722750849468205600421097, 4.09517336699224355643020182410, 5.87845938002171052402368009952, 6.44617776957637679148473979356, 7.48969769720563743985214897235, 8.305375748960281157955725839033, 9.804862445508370487002123458314, 10.66096253843872665347137863176, 10.94460665802989277271206181916

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