Properties

Label 2-546-91.24-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.606 - 0.795i$
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  + (−0.965 + 0.258i)2-s + i·3-s + (0.866 − 0.499i)4-s + (3.90 + 1.04i)5-s + (−0.258 − 0.965i)6-s + (−1.71 + 2.01i)7-s + (−0.707 + 0.707i)8-s − 9-s − 4.04·10-s + (−3.58 + 3.58i)11-s + (0.499 + 0.866i)12-s + (−2.15 + 2.88i)13-s + (1.13 − 2.39i)14-s + (−1.04 + 3.90i)15-s + (0.500 − 0.866i)16-s + (−2.48 − 4.29i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (1.74 + 0.468i)5-s + (−0.105 − 0.394i)6-s + (−0.648 + 0.761i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 1.27·10-s + (−1.08 + 1.08i)11-s + (0.144 + 0.249i)12-s + (−0.598 + 0.801i)13-s + (0.303 − 0.638i)14-s + (−0.270 + 1.00i)15-s + (0.125 − 0.216i)16-s + (−0.601 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478397 + 0.966557i\)
\(L(\frac12)\) \(\approx\) \(0.478397 + 0.966557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 - iT \)
7 \( 1 + (1.71 - 2.01i)T \)
13 \( 1 + (2.15 - 2.88i)T \)
good5 \( 1 + (-3.90 - 1.04i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.58 - 3.58i)T - 11iT^{2} \)
17 \( 1 + (2.48 + 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.02 + 1.02i)T - 19iT^{2} \)
23 \( 1 + (-4.70 - 2.71i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.24 + 7.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.06 - 3.97i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.202 + 0.754i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.46 + 0.927i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.45 - 4.88i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.761 - 2.84i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.08 + 3.61i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.242 + 0.903i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 + (-6.50 - 6.50i)T + 67iT^{2} \)
71 \( 1 + (4.28 - 1.14i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-12.8 + 3.43i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.32 - 2.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.02 + 1.02i)T - 83iT^{2} \)
89 \( 1 + (-6.18 + 1.65i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.62 - 9.79i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.75768091025179779951156774923, −9.834843560473859790469751472633, −9.558636003263084979738803902481, −8.960818361829700080481312926101, −7.36462079638387283944740059918, −6.63047467072033467334311387053, −5.60425876048408855071127828698, −4.90737151258642547460747241004, −2.72759871743469685894322362587, −2.19773595195751152545412144984, 0.73586734529821199158976607717, 2.13541887598664600908119463117, 3.19191211632912126219469615602, 5.20663526349887678660647364949, 5.99381998423224723177235525192, 6.84177329433842802400352694229, 7.939189411687148569372057323423, 8.824156579762811028448012151136, 9.615904428052340455435046635967, 10.59661530567058771973855048248

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