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Properties

Label 2-546-7.2-c3-0-18
Degree $2$
Conductor $546$
Sign $-0.540 - 0.841i$
Analytic cond. $32.2150$
Root an. cond. $5.67582$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (1.23 − 2.13i)5-s − 6·6-s + (15.8 + 9.51i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (2.46 + 4.27i)10-s + (−27.0 − 46.9i)11-s + (6.00 − 10.3i)12-s − 13·13-s + (−32.3 + 18.0i)14-s + 7.40·15-s + (−8 + 13.8i)16-s + (52.3 + 90.5i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.110 − 0.191i)5-s − 0.408·6-s + (0.858 + 0.513i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.0780 + 0.135i)10-s + (−0.742 − 1.28i)11-s + (0.144 − 0.249i)12-s − 0.277·13-s + (−0.617 + 0.343i)14-s + 0.127·15-s + (−0.125 + 0.216i)16-s + (0.746 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.671020552\)
\(L(\frac12)\) \(\approx\) \(1.671020552\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (-15.8 - 9.51i)T \)
13 \( 1 + 13T \)
good5 \( 1 + (-1.23 + 2.13i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (27.0 + 46.9i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-52.3 - 90.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-18.6 + 32.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (55.6 - 96.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 61.8T + 2.43e4T^{2} \)
31 \( 1 + (-11.8 - 20.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-2.14 + 3.71i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 4.37T + 6.89e4T^{2} \)
43 \( 1 - 345.T + 7.95e4T^{2} \)
47 \( 1 + (-11.9 + 20.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-284. - 493. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (103. + 180. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (253. - 438. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-473. - 820. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 559.T + 3.57e5T^{2} \)
73 \( 1 + (-441. - 765. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (253. - 438. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 698.T + 5.71e5T^{2} \)
89 \( 1 + (-229. + 397. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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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

−10.63186900063037230075306296395, −9.666155952435324949194673223841, −8.692243576390551929106308162874, −8.229290722333409021513576159046, −7.36166642372335664492226418081, −5.73519134807968132817367533337, −5.47351602061196089562959213369, −4.16502741560340279484457431292, −2.81707290570393617346487850904, −1.25896518192302090569515556907, 0.59854301879286792424997657909, 1.94090694709908756970021703729, 2.81234314142328590794686614488, 4.31648356787154898679955708987, 5.18113773136409999131752765872, 6.78667567595017572699831855996, 7.63236947507932600809507388165, 8.129730782126370743021669326604, 9.380838200252937114983406712834, 10.15210740298078895180630098145

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