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Properties

Label 2-546-273.254-c1-0-37
Degree $2$
Conductor $546$
Sign $-0.775 + 0.631i$
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  + (0.965 + 0.258i)2-s + (−0.287 − 1.70i)3-s + (0.866 + 0.499i)4-s + (−0.571 − 2.13i)5-s + (0.163 − 1.72i)6-s + (−2.64 − 0.112i)7-s + (0.707 + 0.707i)8-s + (−2.83 + 0.983i)9-s − 2.20i·10-s + (−1.49 − 1.49i)11-s + (0.604 − 1.62i)12-s + (−3.42 − 1.11i)13-s + (−2.52 − 0.792i)14-s + (−3.48 + 1.59i)15-s + (0.500 + 0.866i)16-s + (2.09 − 3.62i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.166 − 0.986i)3-s + (0.433 + 0.249i)4-s + (−0.255 − 0.954i)5-s + (0.0668 − 0.703i)6-s + (−0.999 − 0.0424i)7-s + (0.249 + 0.249i)8-s + (−0.944 + 0.327i)9-s − 0.698i·10-s + (−0.450 − 0.450i)11-s + (0.174 − 0.468i)12-s + (−0.950 − 0.310i)13-s + (−0.674 − 0.211i)14-s + (−0.898 + 0.410i)15-s + (0.125 + 0.216i)16-s + (0.508 − 0.879i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415761 - 1.16791i\)
\(L(\frac12)\) \(\approx\) \(0.415761 - 1.16791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.287 + 1.70i)T \)
7 \( 1 + (2.64 + 0.112i)T \)
13 \( 1 + (3.42 + 1.11i)T \)
good5 \( 1 + (0.571 + 2.13i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.49 + 1.49i)T + 11iT^{2} \)
17 \( 1 + (-2.09 + 3.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.337 - 0.337i)T + 19iT^{2} \)
23 \( 1 + (-2.41 - 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.814 + 0.470i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.14 + 4.28i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (8.78 + 2.35i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.42 + 9.06i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.32 + 1.34i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.06 + 1.08i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.82 + 2.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 4.04T + 61T^{2} \)
67 \( 1 + (6.00 + 6.00i)T + 67iT^{2} \)
71 \( 1 + (-5.98 - 1.60i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.72 - 1.80i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.77 - 3.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.36 - 7.36i)T - 83iT^{2} \)
89 \( 1 + (-4.08 + 15.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.62 - 17.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.62698249818059216023590507900, −9.458744244229610395497274838439, −8.497095020159321851635200821631, −7.49386552920427643865877736398, −6.91264185055197303407128402888, −5.60863216234983100846315258608, −5.16866591586723688926990882406, −3.57803889017906222716480776872, −2.45542957355745359651796096839, −0.55522356956779053953755637923, 2.66366697585176697631881618105, 3.36759384758967450192027240169, 4.44295627528742109234732861338, 5.44927146283431423600663217447, 6.51695618239172265735471873581, 7.21897028326620128456859027146, 8.649087500623305147304706410931, 9.842491651932292947869673102672, 10.29990063254552050663598841114, 10.98280442528516859986387393060

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