Properties

Label 2-546-91.25-c1-0-18
Degree $2$
Conductor $546$
Sign $-0.102 + 0.994i$
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 + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.935 − 0.540i)5-s − 0.999i·6-s + (1.55 − 2.14i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.540 − 0.935i)10-s + (−4.99 − 2.88i)11-s + (−0.499 − 0.866i)12-s + (0.235 + 3.59i)13-s + (0.275 − 2.63i)14-s − 1.08i·15-s + (−0.5 − 0.866i)16-s + (−1.54 + 2.68i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (0.418 − 0.241i)5-s − 0.408i·6-s + (0.587 − 0.809i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.170 − 0.295i)10-s + (−1.50 − 0.870i)11-s + (−0.144 − 0.249i)12-s + (0.0651 + 0.997i)13-s + (0.0735 − 0.703i)14-s − 0.278i·15-s + (−0.125 − 0.216i)16-s + (−0.375 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.102 + 0.994i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.102 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56667 - 1.73582i\)
\(L(\frac12)\) \(\approx\) \(1.56667 - 1.73582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.55 + 2.14i)T \)
13 \( 1 + (-0.235 - 3.59i)T \)
good5 \( 1 + (-0.935 + 0.540i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.99 + 2.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.54 - 2.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.93 + 3.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.46 - 4.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.05T + 29T^{2} \)
31 \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.06 - 1.19i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.58iT - 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 + (-0.297 + 0.171i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.09 - 8.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.94 + 5.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.01 - 3.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (-6.41 - 3.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.98 - 3.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (-8.96 + 5.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.93iT - 97T^{2} \)
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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

−10.92084534824692324817812556201, −9.764670019513119751860078130922, −8.811250776175027724561896348143, −7.73698446926511624712727670050, −7.00745720203885925699575331212, −5.73240821421144742825767049834, −4.97005138751842250261458940750, −3.70996227894735986395099268066, −2.50475907188947679748914969957, −1.19107364346317785511268681745, 2.34480543328714188314843478289, 3.09676635596626435943534369984, 4.82309137408653268731558087546, 5.18532493604721326272236287562, 6.25429875854973339914471550645, 7.65222879763261511487380641722, 8.107734653751344824675577278048, 9.311190592059085702397162604003, 10.23151900818252103923407773273, 10.90960410946971718080398675311

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