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Properties

Label 2-546-273.44-c1-0-22
Degree $2$
Conductor $546$
Sign $0.913 - 0.406i$
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 + (1.26 + 1.18i)3-s + (0.866 − 0.499i)4-s + (1.89 − 0.508i)5-s + (1.52 + 0.821i)6-s + (1.71 + 2.01i)7-s + (0.707 − 0.707i)8-s + (0.177 + 2.99i)9-s + (1.70 − 0.982i)10-s + (−2.87 − 0.771i)11-s + (1.68 + 0.398i)12-s + (−3.35 + 1.31i)13-s + (2.17 + 1.50i)14-s + (2.99 + 1.61i)15-s + (0.500 − 0.866i)16-s + (−1.01 − 1.76i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.727 + 0.685i)3-s + (0.433 − 0.249i)4-s + (0.848 − 0.227i)5-s + (0.622 + 0.335i)6-s + (0.647 + 0.761i)7-s + (0.249 − 0.249i)8-s + (0.0590 + 0.998i)9-s + (0.538 − 0.310i)10-s + (−0.867 − 0.232i)11-s + (0.486 + 0.115i)12-s + (−0.930 + 0.365i)13-s + (0.581 + 0.401i)14-s + (0.773 + 0.416i)15-s + (0.125 − 0.216i)16-s + (−0.247 − 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.913 - 0.406i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93678 + 0.623681i\)
\(L(\frac12)\) \(\approx\) \(2.93678 + 0.623681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 - 1.18i)T \)
7 \( 1 + (-1.71 - 2.01i)T \)
13 \( 1 + (3.35 - 1.31i)T \)
good5 \( 1 + (-1.89 + 0.508i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.87 + 0.771i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.01 + 1.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.12 + 4.21i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.60 + 2.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.15iT - 29T^{2} \)
31 \( 1 + (2.95 + 0.791i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.92 + 1.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.22 - 7.22i)T + 41iT^{2} \)
43 \( 1 + 1.92iT - 43T^{2} \)
47 \( 1 + (0.203 + 0.758i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.77 + 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.58 - 0.692i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.72 + 4.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.655 + 0.175i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.64 + 7.64i)T + 71iT^{2} \)
73 \( 1 + (3.05 - 11.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.89 - 8.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.83 + 4.83i)T + 83iT^{2} \)
89 \( 1 + (-3.24 - 12.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \)
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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

−10.92429868857143093862279901673, −9.864903240370934470017609020169, −9.284603323034279161891699276119, −8.327122956231998586599922132182, −7.31969533490032851122107678190, −5.87897160613052271483283479928, −5.06353610892301210888860189277, −4.40161619159105822491250904075, −2.68702719413025113958247170586, −2.22054749830511645103868937873, 1.70289533575580512839003698050, 2.69171359632530055954478226585, 3.96474424235024146940677719837, 5.21266255952386988466571200035, 6.14539027075791572586939006084, 7.33929225223521293921053925118, 7.67128498020493596229474956430, 8.816684459980961956507511821554, 10.02661073470197443403806862970, 10.62916223902876445945955823443

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