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Properties

Label 2-546-91.23-c1-0-9
Degree $2$
Conductor $546$
Sign $0.642 + 0.766i$
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 + (2.27 + 1.31i)5-s + (0.866 + 0.5i)6-s + (−0.116 − 2.64i)7-s + i·8-s + (−0.499 − 0.866i)9-s + (1.31 − 2.27i)10-s + (−4.33 − 2.50i)11-s + (0.5 − 0.866i)12-s + (3.59 − 0.285i)13-s + (−2.64 + 0.116i)14-s + (−2.27 + 1.31i)15-s + 16-s + 6.89·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 + 0.499i)3-s − 0.5·4-s + (1.01 + 0.586i)5-s + (0.353 + 0.204i)6-s + (−0.0441 − 0.999i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.414 − 0.718i)10-s + (−1.30 − 0.755i)11-s + (0.144 − 0.249i)12-s + (0.996 − 0.0790i)13-s + (−0.706 + 0.0312i)14-s + (−0.586 + 0.338i)15-s + 0.250·16-s + 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33878 - 0.624816i\)
\(L(\frac12)\) \(\approx\) \(1.33878 - 0.624816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.116 + 2.64i)T \)
13 \( 1 + (-3.59 + 0.285i)T \)
good5 \( 1 + (-2.27 - 1.31i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.33 + 2.50i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.89T + 17T^{2} \)
19 \( 1 + (-2.62 + 1.51i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.99T + 23T^{2} \)
29 \( 1 + (-2.18 - 3.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.93 - 2.84i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.28iT - 37T^{2} \)
41 \( 1 + (-4.95 + 2.85i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.44 + 11.1i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.499 + 0.288i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.49 + 6.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + (7.15 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.14 + 1.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.82 - 1.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.20 - 2.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.69 - 6.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.16iT - 83T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 + (12.1 + 7.03i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.72486722536422056713050541901, −10.12295401987852817803917574257, −9.269645997506278603430188419381, −8.131636448820723386281268193270, −7.02764848387907942618589527872, −5.75506080147093791928801486459, −5.16812781644643557356528347444, −3.61370423069655341910519167790, −2.89923898024511571415719032604, −1.07533182708485559163364297927, 1.42741439729113131302727210259, 2.87638622261926247715375741604, 4.80008017017825331393395527737, 5.80993887528286849840185421496, 5.85113420161917043059323884303, 7.44641440448260491410581285228, 8.070347590247387006291974655977, 9.194547858152413777135761877629, 9.741391383377679287385904320351, 10.84207149246907477972113400663

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