Properties

Label 2-546-273.2-c1-0-28
Degree $2$
Conductor $546$
Sign $0.233 + 0.972i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (−1.36 + 1.06i)3-s + 1.00i·4-s + (−0.465 + 0.124i)5-s + (−1.71 − 0.213i)6-s + (−1.66 − 2.05i)7-s + (−0.707 + 0.707i)8-s + (0.733 − 2.90i)9-s + (−0.416 − 0.240i)10-s + (−1.51 − 5.66i)11-s + (−1.06 − 1.36i)12-s + (−3.55 + 0.592i)13-s + (0.279 − 2.63i)14-s + (0.502 − 0.665i)15-s − 1.00·16-s + 0.466·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.788 + 0.614i)3-s + 0.500i·4-s + (−0.207 + 0.0557i)5-s + (−0.701 − 0.0870i)6-s + (−0.628 − 0.777i)7-s + (−0.250 + 0.250i)8-s + (0.244 − 0.969i)9-s + (−0.131 − 0.0761i)10-s + (−0.457 − 1.70i)11-s + (−0.307 − 0.394i)12-s + (−0.986 + 0.164i)13-s + (0.0746 − 0.703i)14-s + (0.129 − 0.171i)15-s − 0.250·16-s + 0.113·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435722 - 0.343646i\)
\(L(\frac12)\) \(\approx\) \(0.435722 - 0.343646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.36 - 1.06i)T \)
7 \( 1 + (1.66 + 2.05i)T \)
13 \( 1 + (3.55 - 0.592i)T \)
good5 \( 1 + (0.465 - 0.124i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.51 + 5.66i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.466T + 17T^{2} \)
19 \( 1 + (-1.43 - 0.385i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.870T + 23T^{2} \)
29 \( 1 + (-7.17 + 4.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.20 + 0.859i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.74 + 3.74i)T + 37iT^{2} \)
41 \( 1 + (-0.926 + 3.45i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.93 + 5.15i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.59 + 5.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (7.10 - 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.57 - 1.57i)T - 59iT^{2} \)
61 \( 1 + (-4.06 - 7.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.66 + 6.20i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.26 + 1.41i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.27 - 12.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + (-4.10 + 4.10i)T - 89iT^{2} \)
97 \( 1 + (-1.48 - 5.54i)T + (-84.0 + 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.63385867027612425588452931973, −9.916175272572396705334410888259, −8.856197694448354992596684111092, −7.70290362414094334415635789767, −6.79685377003824906794049336102, −5.90013646455304097972289904568, −5.10579117651980035044798561986, −3.94849616112230699673207225928, −3.15687297765892924473714439843, −0.28643068668777670021278831607, 1.84696266407075795737591964886, 2.91898824704055046329407274205, 4.68958934489262326179358873691, 5.18799068325387099300841678505, 6.40875410773107875876363250905, 7.14532563881997715601647945100, 8.171807789580702385560410534008, 9.728378923754285929285251942776, 10.03837576685369773034055521299, 11.23860919383286036799338309851

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