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Properties

Label 2-546-91.4-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.997 + 0.0742i$
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  − i·2-s + (−0.5 − 0.866i)3-s − 4-s + (−0.910 + 0.525i)5-s + (−0.866 + 0.5i)6-s + (0.963 − 2.46i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (0.525 + 0.910i)10-s + (5.68 − 3.28i)11-s + (0.5 + 0.866i)12-s + (−3.07 − 1.88i)13-s + (−2.46 − 0.963i)14-s + (0.910 + 0.525i)15-s + 16-s − 4.62·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (−0.407 + 0.234i)5-s + (−0.353 + 0.204i)6-s + (0.364 − 0.931i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.166 + 0.287i)10-s + (1.71 − 0.989i)11-s + (0.144 + 0.249i)12-s + (−0.852 − 0.522i)13-s + (−0.658 − 0.257i)14-s + (0.234 + 0.135i)15-s + 0.250·16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.997 + 0.0742i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.997 + 0.0742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0327084 - 0.880398i\)
\(L(\frac12)\) \(\approx\) \(0.0327084 - 0.880398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.963 + 2.46i)T \)
13 \( 1 + (3.07 + 1.88i)T \)
good5 \( 1 + (0.910 - 0.525i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.68 + 3.28i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.62T + 17T^{2} \)
19 \( 1 + (2.59 + 1.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.19T + 23T^{2} \)
29 \( 1 + (1.12 - 1.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.58 + 4.38i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.64iT - 37T^{2} \)
41 \( 1 + (-6.65 - 3.84i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.696 - 1.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.01 + 5.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.40 + 5.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.49iT - 59T^{2} \)
61 \( 1 + (-4.95 + 8.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.78 - 4.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.13 + 4.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.18 + 4.72i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.65 - 2.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.79iT - 83T^{2} \)
89 \( 1 + 4.35iT - 89T^{2} \)
97 \( 1 + (-5.83 + 3.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.79326585899453044146997337726, −9.533133040909278620237775573667, −8.698320944894754958022585925083, −7.61818054023046218244129835880, −6.84899663862304756915762003132, −5.74180305556436810527679322771, −4.33185087967535470091848974683, −3.61965720849135941671739904451, −2.00786501973952629215729791760, −0.52860504946692225218921330970, 2.04741257430271718297243676975, 4.13434694196364276554696492965, 4.50189668981477758201028819713, 5.77365869552437494422636968859, 6.63825709800294577459774557765, 7.56034191184668504076611275185, 8.890822431851866156941190058367, 9.102419309865414467551208454317, 10.15624961770927414596590336110, 11.40699929200866039615097066964

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