Properties

Label 2-546-91.81-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.411 - 0.911i$
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 − 3-s + (−0.499 − 0.866i)4-s + (1.14 + 1.98i)5-s + (0.5 − 0.866i)6-s + (−1.12 − 2.39i)7-s + 0.999·8-s + 9-s − 2.29·10-s + 0.878·11-s + (0.499 + 0.866i)12-s + (−0.786 + 3.51i)13-s + (2.63 + 0.222i)14-s + (−1.14 − 1.98i)15-s + (−0.5 + 0.866i)16-s + (3.20 + 5.54i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.512 + 0.887i)5-s + (0.204 − 0.353i)6-s + (−0.425 − 0.904i)7-s + 0.353·8-s + 0.333·9-s − 0.724·10-s + 0.264·11-s + (0.144 + 0.249i)12-s + (−0.218 + 0.975i)13-s + (0.704 + 0.0593i)14-s + (−0.295 − 0.512i)15-s + (−0.125 + 0.216i)16-s + (0.777 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.411 - 0.911i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488239 + 0.756351i\)
\(L(\frac12)\) \(\approx\) \(0.488239 + 0.756351i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (1.12 + 2.39i)T \)
13 \( 1 + (0.786 - 3.51i)T \)
good5 \( 1 + (-1.14 - 1.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.878T + 11T^{2} \)
17 \( 1 + (-3.20 - 5.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.50T + 19T^{2} \)
23 \( 1 + (0.658 - 1.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.669 + 1.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.94 - 3.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.69 - 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.80 - 3.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.95 - 8.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.188 + 0.327i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 - 2.11i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.98 - 5.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.81T + 61T^{2} \)
67 \( 1 - 9.75T + 67T^{2} \)
71 \( 1 + (-1.02 + 1.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.432 + 0.749i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + (-6.41 + 11.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.40 + 7.62i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.81658855692938055253215925723, −10.11682514950962295841908483275, −9.605779163495130796922181261560, −8.279806933108269160065018036667, −7.20377968892585560700047576749, −6.58555853761653970799052031358, −5.93162342332336860876797624784, −4.60459595486079229132379207505, −3.42179694352292648995554408399, −1.51854599648780713647240269983, 0.67215294790474298877177235815, 2.23961935161897979127902966087, 3.56167057514259213955537003699, 5.26173157642786761953107896722, 5.43425213835938749418423109167, 6.91383202655041222235093222516, 8.043434044090107013504291384423, 9.116097772645567371588763968426, 9.557776348973200693256219138972, 10.43714694777151284240575854867

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