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Properties

Label 2-546-91.9-c1-0-4
Degree $2$
Conductor $546$
Sign $0.703 - 0.710i$
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 + (0.5 + 0.866i)6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + 9-s − 2·11-s + (0.499 − 0.866i)12-s + (3.5 + 0.866i)13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2 + 3.46i)17-s + (−0.5 − 0.866i)18-s + 19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + 0.333·9-s − 0.603·11-s + (0.144 − 0.249i)12-s + (0.970 + 0.240i)13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.117 − 0.204i)18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588760 + 0.245434i\)
\(L(\frac12)\) \(\approx\) \(0.588760 + 0.245434i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 + 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96505493948153775505026564181, −10.14656499049722708863401473638, −9.345631333390841040573224792489, −8.426439441411713952972488830057, −7.27417136595214622397978430678, −6.40560230462342402064102938847, −5.33585081303788775726538513519, −4.02128339776417857138440775527, −3.06626493886413318983046894820, −1.33290728139460089035779235365, 0.48937629211104165898280120423, 2.64311705229149036855107725981, 4.17572968269952603645565269157, 5.39310197254778900217970881164, 6.16760027991570184922808579047, 6.92525076046546996465140001714, 7.967480470006310069205083854215, 8.971174371970919606599974057965, 9.716450555996557231814114486217, 10.63808156399107579626554040805

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