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Properties

Label 2-546-13.9-c3-0-27
Degree $2$
Conductor $546$
Sign $0.542 + 0.839i$
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 + 3.20·5-s + (3 + 5.19i)6-s + (3.5 + 6.06i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−3.20 + 5.55i)10-s + (1.21 − 2.09i)11-s − 12·12-s + (9.54 − 45.8i)13-s − 14·14-s + (4.81 − 8.33i)15-s + (−8 + 13.8i)16-s + (4.86 + 8.43i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.286·5-s + (0.204 + 0.353i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.101 + 0.175i)10-s + (0.0331 − 0.0574i)11-s − 0.288·12-s + (0.203 − 0.979i)13-s − 0.267·14-s + (0.0828 − 0.143i)15-s + (−0.125 + 0.216i)16-s + (0.0694 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.542 + 0.839i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ 0.542 + 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.611912754\)
\(L(\frac12)\) \(\approx\) \(1.611912754\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 - 6.06i)T \)
13 \( 1 + (-9.54 + 45.8i)T \)
good5 \( 1 - 3.20T + 125T^{2} \)
11 \( 1 + (-1.21 + 2.09i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-4.86 - 8.43i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.47 - 6.01i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-77.6 + 134. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (78.5 - 136. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 190.T + 2.97e4T^{2} \)
37 \( 1 + (31.1 - 53.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-122. + 211. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (81.1 + 140. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 217.T + 1.03e5T^{2} \)
53 \( 1 - 625.T + 1.48e5T^{2} \)
59 \( 1 + (253. + 439. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (431. + 746. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-274. + 474. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (368. + 639. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 554.T + 3.89e5T^{2} \)
79 \( 1 + 765.T + 4.93e5T^{2} \)
83 \( 1 - 341.T + 5.71e5T^{2} \)
89 \( 1 + (-778. + 1.34e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-827. - 1.43e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.15085050278197898876527998316, −9.131885976780243852392580640865, −8.389041888908736599771174506544, −7.67310841756015907061386950162, −6.64089677606471395687445130423, −5.83186205619854394582170772757, −4.85674872207651744813116462295, −3.29566552071219643663133598359, −1.96427173214913202445784805670, −0.56791093315740008451264444308, 1.28345929421616659572954803621, 2.50965863717192055393047417189, 3.75573475127804296852555582594, 4.58842987389425698578986658266, 5.83571993857163245727915420863, 7.13086475691983437627208522064, 8.041645550758917090084443581162, 9.054577269360590250413854688439, 9.662073116828454956817247367829, 10.39467550575627509596076561797

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