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Properties

Label 2-546-91.30-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.570 + 0.821i$
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  + (0.866 − 0.5i)2-s − 3-s + (0.499 − 0.866i)4-s + (−0.152 − 0.0882i)5-s + (−0.866 + 0.5i)6-s + (−0.623 − 2.57i)7-s − 0.999i·8-s + 9-s − 0.176·10-s − 2.92i·11-s + (−0.499 + 0.866i)12-s + (−2.87 + 2.18i)13-s + (−1.82 − 1.91i)14-s + (0.152 + 0.0882i)15-s + (−0.5 − 0.866i)16-s + (−0.536 + 0.928i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s − 0.577·3-s + (0.249 − 0.433i)4-s + (−0.0683 − 0.0394i)5-s + (−0.353 + 0.204i)6-s + (−0.235 − 0.971i)7-s − 0.353i·8-s + 0.333·9-s − 0.0558·10-s − 0.882i·11-s + (−0.144 + 0.249i)12-s + (−0.796 + 0.604i)13-s + (−0.487 − 0.511i)14-s + (0.0394 + 0.0227i)15-s + (−0.125 − 0.216i)16-s + (−0.130 + 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.570 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605860 - 1.15843i\)
\(L(\frac12)\) \(\approx\) \(0.605860 - 1.15843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + T \)
7 \( 1 + (0.623 + 2.57i)T \)
13 \( 1 + (2.87 - 2.18i)T \)
good5 \( 1 + (0.152 + 0.0882i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.92iT - 11T^{2} \)
17 \( 1 + (0.536 - 0.928i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 + (0.966 + 1.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.880 + 1.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.771 + 0.445i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.26 + 3.61i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.65 + 2.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.42 + 4.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.56 - 3.79i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.98 - 6.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.3 - 7.15i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.40T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + (-11.6 + 6.71i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.20 - 5.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.86 + 6.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.49iT - 83T^{2} \)
89 \( 1 + (-4.92 + 2.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.29 + 5.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.63102322963271025967837053898, −9.957478692792577692532645853228, −8.862621727501180591248747482081, −7.53137649657736803845524223364, −6.70298844079820713518573226078, −5.83550521902106983522781965236, −4.64013691357490321787001279181, −3.93742618282544470760272612965, −2.49702489626518820096562935514, −0.65134728135642011864161717466, 2.10900590845158351613828712453, 3.45629499272390317693533151981, 4.80037773858005993546791661465, 5.49569921244882053694420915362, 6.39095862673419548890885538925, 7.37138017995012499848698275462, 8.214646582146943588630470799047, 9.520225010471137313001573333702, 10.15371988119601113715466185516, 11.42918562954515361498670028830

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