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Properties

Label 2-546-273.101-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.568 - 0.822i$
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 + (−1.23 − 1.21i)3-s + (−0.499 − 0.866i)4-s + (0.567 − 0.327i)5-s + (1.66 − 0.459i)6-s + (−2.19 − 1.46i)7-s + 0.999·8-s + (0.0419 + 2.99i)9-s + 0.655i·10-s − 3.09·11-s + (−0.436 + 1.67i)12-s + (−3.50 − 0.844i)13-s + (2.37 − 1.17i)14-s + (−1.09 − 0.286i)15-s + (−0.5 + 0.866i)16-s + (3.51 + 6.08i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.712 − 0.702i)3-s + (−0.249 − 0.433i)4-s + (0.253 − 0.146i)5-s + (0.681 − 0.187i)6-s + (−0.831 − 0.555i)7-s + 0.353·8-s + (0.0139 + 0.999i)9-s + 0.207i·10-s − 0.932·11-s + (−0.126 + 0.483i)12-s + (−0.972 − 0.234i)13-s + (0.634 − 0.312i)14-s + (−0.283 − 0.0738i)15-s + (−0.125 + 0.216i)16-s + (0.851 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.568 - 0.822i$
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: \((2,\ 546,\ (\ :1/2),\ -0.568 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161787 + 0.308413i\)
\(L(\frac12)\) \(\approx\) \(0.161787 + 0.308413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.21i)T \)
7 \( 1 + (2.19 + 1.46i)T \)
13 \( 1 + (3.50 + 0.844i)T \)
good5 \( 1 + (-0.567 + 0.327i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
17 \( 1 + (-3.51 - 6.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.51T + 19T^{2} \)
23 \( 1 + (-1.24 - 0.717i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.89 - 2.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.26 - 3.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.56 - 1.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.52 - 4.34i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0380 - 0.0658i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.04 - 4.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.68 + 5.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.13 + 3.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 15.3iT - 61T^{2} \)
67 \( 1 + 3.99iT - 67T^{2} \)
71 \( 1 + (-0.469 + 0.813i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.44 + 9.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.40 + 2.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.24iT - 83T^{2} \)
89 \( 1 + (-9.46 - 5.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.92056719796696310893021977087, −10.12086099907020228694784950357, −9.533419377564105348092284791959, −8.026550224784109732677305301408, −7.52002453016838007256831956450, −6.65841183645037864731743678296, −5.64126543455967454981742329910, −5.05232440678584808511837437641, −3.26239365962614936410680245955, −1.44058649981977383819333185828, 0.25219903608474628271643863694, 2.55249541931492743998501859134, 3.46450491281580775336019306393, 4.98497694297455620180866620260, 5.56978549285103678645847083108, 6.87755388716358015821546240128, 7.84692573909547608717548757301, 9.380109783309799911407177392758, 9.645374400037846699126310265019, 10.28140349313648532863160820576

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