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Properties

Label 2-546-273.101-c1-0-20
Degree $2$
Conductor $546$
Sign $0.707 + 0.706i$
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 + (1.72 − 0.0896i)3-s + (−0.499 − 0.866i)4-s + (−3.27 + 1.89i)5-s + (0.787 − 1.54i)6-s + (2.49 − 0.889i)7-s − 0.999·8-s + (2.98 − 0.310i)9-s + 3.78i·10-s + 5.80·11-s + (−0.942 − 1.45i)12-s + (0.879 − 3.49i)13-s + (0.475 − 2.60i)14-s + (−5.50 + 3.56i)15-s + (−0.5 + 0.866i)16-s + (2.26 + 3.93i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.998 − 0.0517i)3-s + (−0.249 − 0.433i)4-s + (−1.46 + 0.846i)5-s + (0.321 − 0.629i)6-s + (0.941 − 0.336i)7-s − 0.353·8-s + (0.994 − 0.103i)9-s + 1.19i·10-s + 1.75·11-s + (−0.272 − 0.419i)12-s + (0.243 − 0.969i)13-s + (0.126 − 0.695i)14-s + (−1.42 + 0.921i)15-s + (−0.125 + 0.216i)16-s + (0.550 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04375 - 0.845933i\)
\(L(\frac12)\) \(\approx\) \(2.04375 - 0.845933i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.0896i)T \)
7 \( 1 + (-2.49 + 0.889i)T \)
13 \( 1 + (-0.879 + 3.49i)T \)
good5 \( 1 + (3.27 - 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 5.80T + 11T^{2} \)
17 \( 1 + (-2.26 - 3.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.85T + 19T^{2} \)
23 \( 1 + (4.19 + 2.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.20 + 1.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.824 + 1.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.865 + 0.499i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.48 - 2.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.445 - 0.772i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.23 - 3.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.41 - 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.67 - 5.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.64iT - 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + (6.77 - 11.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.19 - 3.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.40 + 14.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (1.89 + 1.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.30 - 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.69518160294095137067612903924, −10.11515585724642844259236509043, −8.665632545177983571378861788591, −8.157903590079769237400082677042, −7.29002688365558888250042397758, −6.23936682538011783630703923997, −4.26272754643999585636911591295, −3.98451978863710304238253448128, −2.96325843812526754267160952821, −1.40120061775259377616142126848, 1.55833371504027328802685865269, 3.55087331370291464101162800217, 4.22861881570956165446095556347, 4.93231147664247027864309669999, 6.61957410351398315272635514216, 7.48405908052429852897201612169, 8.316310919674792374626526402772, 8.790058981809701043253901087932, 9.540227283457460986343158081656, 11.29299873832242282454677246529

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