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Properties

Label 2-546-21.5-c1-0-22
Degree $2$
Conductor $546$
Sign $-0.720 + 0.693i$
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 + (−1.44 − 0.959i)3-s + (0.499 + 0.866i)4-s + (1.50 − 2.61i)5-s + (0.768 + 1.55i)6-s + (0.237 − 2.63i)7-s − 0.999i·8-s + (1.15 + 2.76i)9-s + (−2.61 + 1.50i)10-s + (4.75 − 2.74i)11-s + (0.110 − 1.72i)12-s − i·13-s + (−1.52 + 2.16i)14-s + (−4.68 + 2.32i)15-s + (−0.5 + 0.866i)16-s + (0.598 + 1.03i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.832 − 0.554i)3-s + (0.249 + 0.433i)4-s + (0.674 − 1.16i)5-s + (0.313 + 0.633i)6-s + (0.0896 − 0.995i)7-s − 0.353i·8-s + (0.385 + 0.922i)9-s + (−0.826 + 0.477i)10-s + (1.43 − 0.827i)11-s + (0.0318 − 0.498i)12-s − 0.277i·13-s + (−0.407 + 0.578i)14-s + (−1.20 + 0.599i)15-s + (−0.125 + 0.216i)16-s + (0.145 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.720 + 0.693i$
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: \((2,\ 546,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369388 - 0.915544i\)
\(L(\frac12)\) \(\approx\) \(0.369388 - 0.915544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 0.959i)T \)
7 \( 1 + (-0.237 + 2.63i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-1.50 + 2.61i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.75 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.598 - 1.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.49 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.22 + 1.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.12iT - 29T^{2} \)
31 \( 1 + (-4.87 + 2.81i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.76 - 8.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.26T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (-3.89 + 6.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.53 - 3.77i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.06 + 3.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.26 - 4.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.97 - 8.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (0.501 - 0.289i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.649 + 1.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (2.75 - 4.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.59iT - 97T^{2} \)
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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

−10.25195201278683321803272304157, −9.861882124149785488260044914071, −8.640606221002961072885355637215, −7.958134862138644425574371365063, −6.78720627263988230627667123821, −5.97873034810441650098435214137, −4.89841304143874854671281706506, −3.68072317929539131426652293675, −1.55954629478040939008639305654, −0.885908916491592433791232432125, 1.75814436496223606450535646991, 3.27836626780908840771102158328, 4.85057210778957849349396030020, 5.82582053176165604370420231174, 6.62609985536327547173779890063, 7.16819958526884129679070281561, 8.795628204372989299978285970411, 9.650394739414866025424287658787, 9.936012641853840610292417687538, 11.12682128621086181092872755729

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