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Properties

Label 2-546-13.10-c3-0-0
Degree $2$
Conductor $546$
Sign $-0.989 + 0.142i$
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  + (−1.73 + i)2-s + (1.5 + 2.59i)3-s + (1.99 − 3.46i)4-s − 11.2i·5-s + (−5.19 − 3i)6-s + (6.06 + 3.5i)7-s + 7.99i·8-s + (−4.5 + 7.79i)9-s + (11.2 + 19.5i)10-s + (−8.19 + 4.73i)11-s + 12·12-s + (−43.7 − 16.8i)13-s − 14·14-s + (29.3 − 16.9i)15-s + (−8 − 13.8i)16-s + (−4.98 + 8.64i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 1.00i·5-s + (−0.353 − 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.356 + 0.618i)10-s + (−0.224 + 0.129i)11-s + 0.288·12-s + (−0.933 − 0.359i)13-s − 0.267·14-s + (0.504 − 0.291i)15-s + (−0.125 − 0.216i)16-s + (−0.0711 + 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1320507065\)
\(L(\frac12)\) \(\approx\) \(0.1320507065\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - i)T \)
3 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (-6.06 - 3.5i)T \)
13 \( 1 + (43.7 + 16.8i)T \)
good5 \( 1 + 11.2iT - 125T^{2} \)
11 \( 1 + (8.19 - 4.73i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (4.98 - 8.64i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-32.1 - 18.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (81.1 + 140. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-44.6 - 77.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 43.4iT - 2.97e4T^{2} \)
37 \( 1 + (70.6 - 40.7i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (373. - 215. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (244. - 423. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 171. iT - 1.03e5T^{2} \)
53 \( 1 - 28.6T + 1.48e5T^{2} \)
59 \( 1 + (242. + 139. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (108. - 188. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (215. - 124. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (994. + 573. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 557. iT - 3.89e5T^{2} \)
79 \( 1 + 144.T + 4.93e5T^{2} \)
83 \( 1 + 1.16e3iT - 5.71e5T^{2} \)
89 \( 1 + (-1.38e3 + 798. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (525. + 303. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.46709110306320697921385209129, −9.917608746117636624936809034389, −8.954664721589987254604223268791, −8.343225196294808162861492541834, −7.59414489044016338001427519669, −6.31322218006251836306396822105, −5.09395995854821225408646177117, −4.59819827019927849968129621050, −2.89573923650594397567675754750, −1.49587021266804336511539847447, 0.04516746058173535025137246311, 1.74352721003272979253393208179, 2.72003919336492938098327616835, 3.76702074153223958647438285576, 5.32527078181160379223353370906, 6.70177048138594560719484596964, 7.29668853176556932665357588219, 8.047423082617484924116537545770, 9.108628128306296366195398351008, 10.01616710894272096510663607599

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