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Properties

Label 2-546-21.5-c1-0-29
Degree $2$
Conductor $546$
Sign $-0.114 + 0.993i$
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  + (0.866 + 0.5i)2-s + (−1.43 − 0.964i)3-s + (0.499 + 0.866i)4-s + (1.85 − 3.22i)5-s + (−0.763 − 1.55i)6-s + (−1.49 − 2.17i)7-s + 0.999i·8-s + (1.14 + 2.77i)9-s + (3.22 − 1.85i)10-s + (−0.553 + 0.319i)11-s + (0.115 − 1.72i)12-s + i·13-s + (−0.209 − 2.63i)14-s + (−5.78 + 2.84i)15-s + (−0.5 + 0.866i)16-s + (−3.63 − 6.30i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.830 − 0.556i)3-s + (0.249 + 0.433i)4-s + (0.831 − 1.44i)5-s + (−0.311 − 0.634i)6-s + (−0.566 − 0.823i)7-s + 0.353i·8-s + (0.380 + 0.924i)9-s + (1.01 − 0.588i)10-s + (−0.166 + 0.0963i)11-s + (0.0334 − 0.498i)12-s + 0.277i·13-s + (−0.0558 − 0.704i)14-s + (−1.49 + 0.733i)15-s + (−0.125 + 0.216i)16-s + (−0.882 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967017 - 1.08461i\)
\(L(\frac12)\) \(\approx\) \(0.967017 - 1.08461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (1.43 + 0.964i)T \)
7 \( 1 + (1.49 + 2.17i)T \)
13 \( 1 - iT \)
good5 \( 1 + (-1.85 + 3.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.553 - 0.319i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.63 + 6.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.07 - 2.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.98iT - 29T^{2} \)
31 \( 1 + (-0.608 + 0.351i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.21 + 9.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 3.57T + 43T^{2} \)
47 \( 1 + (3.53 - 6.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.65 + 3.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.35 + 2.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.0 - 5.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.15 + 1.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (8.28 - 4.78i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.00 - 8.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (-1.95 + 3.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.50iT - 97T^{2} \)
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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

−10.79448188924193923070178301284, −9.567543322506368649902335925512, −8.867006888078821368570907758276, −7.52128272166867501879102777228, −6.78725356214271578069242613181, −5.89334647913189972340274795140, −4.94560082806032371947947697536, −4.36974353303270428916193754713, −2.33822103205239974098225065491, −0.74813317373585295989059501783, 2.18363062815801005025275263852, 3.20985715141306774262700537042, 4.36675554287179407041208667542, 5.77276642074448906790509630612, 6.17982444298823571831786336584, 6.82232905303700824976133670727, 8.595053691995558372753182610455, 9.779387400146463540466376044528, 10.36385088293588296981477812524, 10.91845400898701499445132623359

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