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Properties

Label 2-546-273.17-c1-0-26
Degree $2$
Conductor $546$
Sign $-0.103 + 0.994i$
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 + (0.929 − 1.46i)3-s + 4-s + (1.09 − 0.634i)5-s + (−0.929 + 1.46i)6-s + (0.151 + 2.64i)7-s − 8-s + (−1.27 − 2.71i)9-s + (−1.09 + 0.634i)10-s + (−2.57 − 4.46i)11-s + (0.929 − 1.46i)12-s + (−2.55 − 2.53i)13-s + (−0.151 − 2.64i)14-s + (0.0940 − 2.19i)15-s + 16-s + 5.00·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.536 − 0.843i)3-s + 0.5·4-s + (0.491 − 0.283i)5-s + (−0.379 + 0.596i)6-s + (0.0572 + 0.998i)7-s − 0.353·8-s + (−0.424 − 0.905i)9-s + (−0.347 + 0.200i)10-s + (−0.776 − 1.34i)11-s + (0.268 − 0.421i)12-s + (−0.709 − 0.704i)13-s + (−0.0404 − 0.705i)14-s + (0.0242 − 0.566i)15-s + 0.250·16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.815468 - 0.904965i\)
\(L(\frac12)\) \(\approx\) \(0.815468 - 0.904965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.929 + 1.46i)T \)
7 \( 1 + (-0.151 - 2.64i)T \)
13 \( 1 + (2.55 + 2.53i)T \)
good5 \( 1 + (-1.09 + 0.634i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.57 + 4.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.00T + 17T^{2} \)
19 \( 1 + (-3.30 + 5.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.70iT - 23T^{2} \)
29 \( 1 + (-0.776 - 0.448i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.25 + 7.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.53iT - 37T^{2} \)
41 \( 1 + (4.54 + 2.62i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.72 - 2.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.18 + 2.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.7 - 6.76i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.8iT - 59T^{2} \)
61 \( 1 + (-4.04 - 2.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.0 - 5.78i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.79 + 6.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.210 + 0.365i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.95 + 5.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.22iT - 83T^{2} \)
89 \( 1 - 8.30iT - 89T^{2} \)
97 \( 1 + (-6.21 - 10.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.39718452653087165459571326031, −9.379938722772309837264196599642, −8.842225545098970498460637363592, −7.911982098407174544385887850093, −7.33157073565660917505026780707, −5.79447207725859426724228574398, −5.54010766524957141997618220370, −3.12201844628921577829582952076, −2.44124398323754047505009785899, −0.842962182711559153762137420678, 1.83962792259148983009594309276, 3.08040073379168924738991476830, 4.35966981830562631829751651055, 5.33987787605461645572203271131, 6.81507398925722962775308712669, 7.63517656540109339168113733843, 8.341130506001157149929188411741, 9.725475571651275034028489415515, 10.09926834103527863041914706020, 10.35560772042714329019314783080

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