Properties

Label 2-546-13.9-c1-0-9
Degree $2$
Conductor $546$
Sign $0.0128 + 0.999i$
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 + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)11-s + 0.999·12-s + (2.5 − 2.59i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s − 0.999·18-s + (−2.5 − 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.452 − 0.783i)11-s + 0.288·12-s + (0.693 − 0.720i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s − 0.235·18-s + (−0.573 − 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01854 - 1.00556i\)
\(L(\frac12)\) \(\approx\) \(1.01854 - 1.00556i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-2.5 + 2.59i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16T + 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8 - 13.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.43665446237303050423048562125, −10.17921788330878460074369981724, −8.862927880834299839115338554495, −8.218760440740126732519333153441, −6.57202235152663544342250571000, −5.93282311916951321499209389756, −4.70971873196188448076294038605, −3.82168485186125513243235329820, −2.78155526401293814983481276927, −0.826149236098139701427696371742, 1.69814671503214991208220847581, 3.36605741552866436779474637925, 4.56427042930626092281714506410, 5.62708270514104307497978292971, 6.50643086392239607624689136621, 7.20688672234468133435500442623, 8.206461721416714192763360892477, 9.122340129898367408359917653256, 10.02605263889822091944870710653, 11.27467675444554506627348569976

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