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Properties

Label 2-546-273.38-c1-0-4
Degree $2$
Conductor $546$
Sign $0.399 - 0.916i$
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 + (0.278 − 1.70i)3-s + (−0.499 + 0.866i)4-s + (−2.44 + 1.41i)5-s + (−1.61 + 0.613i)6-s + (2.60 − 0.449i)7-s + 0.999·8-s + (−2.84 − 0.950i)9-s + (2.44 + 1.41i)10-s + (−2.97 + 5.15i)11-s + (1.34 + 1.09i)12-s + (−3.07 − 1.87i)13-s + (−1.69 − 2.03i)14-s + (1.73 + 4.58i)15-s + (−0.5 − 0.866i)16-s + (−2.83 + 4.91i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.160 − 0.987i)3-s + (−0.249 + 0.433i)4-s + (−1.09 + 0.632i)5-s + (−0.661 + 0.250i)6-s + (0.985 − 0.169i)7-s + 0.353·8-s + (−0.948 − 0.316i)9-s + (0.774 + 0.447i)10-s + (−0.897 + 1.55i)11-s + (0.387 + 0.316i)12-s + (−0.854 − 0.520i)13-s + (−0.452 − 0.543i)14-s + (0.448 + 1.18i)15-s + (−0.125 − 0.216i)16-s + (−0.688 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.399 - 0.916i$
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: \((2,\ 546,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373821 + 0.244885i\)
\(L(\frac12)\) \(\approx\) \(0.373821 + 0.244885i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.278 + 1.70i)T \)
7 \( 1 + (-2.60 + 0.449i)T \)
13 \( 1 + (3.07 + 1.87i)T \)
good5 \( 1 + (2.44 - 1.41i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.97 - 5.15i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.972 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.43 + 1.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.0658iT - 29T^{2} \)
31 \( 1 + (4.39 - 7.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.160 - 0.0928i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.04iT - 41T^{2} \)
43 \( 1 - 3.48T + 43T^{2} \)
47 \( 1 + (-6.77 + 3.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.77 - 2.18i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.6 + 6.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.8 - 7.39i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.912 + 0.527i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.16T + 71T^{2} \)
73 \( 1 + (-0.140 + 0.242i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.18 - 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.940iT - 83T^{2} \)
89 \( 1 + (-4.61 + 2.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.60T + 97T^{2} \)
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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

−10.72433437614048649519008907172, −10.61541338582013752720835550252, −9.052642007255430093447561417256, −8.025933287885647208334039623658, −7.55501669590882377594566219991, −6.96826099650934434287680547356, −5.23178845712707986110908611916, −4.11325677366190380580547024011, −2.77172182042785231041440330062, −1.74643597811300720470494317698, 0.27754060041015215149202729688, 2.79305824463858716803649934715, 4.29054170226115972625307316737, 4.91977634883868260666254457702, 5.74961137227403060585783127012, 7.47879986584182932389007915285, 7.993497256610990988748999995236, 8.897820177190380681616042260833, 9.355353976727376667604420012616, 10.82536761750071946911363194267

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