Properties

Label 2-546-273.257-c1-0-23
Degree $2$
Conductor $546$
Sign $-0.190 + 0.981i$
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  − 2-s − 1.73i·3-s + 4-s + (1.5 + 0.866i)5-s + 1.73i·6-s + (−2 + 1.73i)7-s − 8-s − 2.99·9-s + (−1.5 − 0.866i)10-s + (1.5 − 2.59i)11-s − 1.73i·12-s + (−1 − 3.46i)13-s + (2 − 1.73i)14-s + (1.49 − 2.59i)15-s + 16-s + 6·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.999i·3-s + 0.5·4-s + (0.670 + 0.387i)5-s + 0.707i·6-s + (−0.755 + 0.654i)7-s − 0.353·8-s − 0.999·9-s + (−0.474 − 0.273i)10-s + (0.452 − 0.783i)11-s − 0.499i·12-s + (−0.277 − 0.960i)13-s + (0.534 − 0.462i)14-s + (0.387 − 0.670i)15-s + 0.250·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600694 - 0.728719i\)
\(L(\frac12)\) \(\approx\) \(0.600694 - 0.728719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + 1.73iT \)
7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + (1 + 3.46i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + (-7.5 + 4.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 + 0.866i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 0.866i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.5 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.39359045466788249225580203444, −9.690985222429625617996817947794, −8.619731516927761445425842223702, −8.068651083609685334264639554292, −6.74699861389655051675361983466, −6.30414506637898754391352688438, −5.41546509918766883852824224614, −3.09614507219141966408188182846, −2.42481906824766069981659866138, −0.69976527940246971659344596072, 1.62082308577913917822821816140, 3.30249027897215130620269749886, 4.32519276222894217755858515757, 5.56908340711862789279267387091, 6.49388866490464604955116058970, 7.56420399145350531179654223740, 8.722501531574962700068674688707, 9.541157999793297379153966001671, 9.999514572158079832985567225461, 10.54747557438362544908538965807

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