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Properties

Label 2-546-13.3-c3-0-30
Degree $2$
Conductor $546$
Sign $0.859 + 0.511i$
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 + 22·5-s + (3 − 5.19i)6-s + (−3.5 + 6.06i)7-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−22 − 38.1i)10-s + (−8 − 13.8i)11-s − 12·12-s + (45.5 − 11.2i)13-s + 14·14-s + (33 + 57.1i)15-s + (−8 − 13.8i)16-s + (49.5 − 85.7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.96·5-s + (0.204 − 0.353i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.695 − 1.20i)10-s + (−0.219 − 0.379i)11-s − 0.288·12-s + (0.970 − 0.240i)13-s + 0.267·14-s + (0.568 + 0.983i)15-s + (−0.125 − 0.216i)16-s + (0.706 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.859 + 0.511i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.674331893\)
\(L(\frac12)\) \(\approx\) \(2.674331893\)
\(L(\frac{5}{2})\) 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 \( 1 + (-45.5 + 11.2i)T \)
good5 \( 1 - 22T + 125T^{2} \)
11 \( 1 + (8 + 13.8i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-49.5 + 85.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-11 + 19.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (76.5 + 132. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (111 + 192. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 91T + 2.97e4T^{2} \)
37 \( 1 + (-133 - 230. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-189 - 327. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (42.5 - 73.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 262T + 1.03e5T^{2} \)
53 \( 1 - 371T + 1.48e5T^{2} \)
59 \( 1 + (257.5 - 446. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (241.5 - 418. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-77.5 - 134. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-424.5 + 735. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 284T + 3.89e5T^{2} \)
79 \( 1 + 116T + 4.93e5T^{2} \)
83 \( 1 - 323T + 5.71e5T^{2} \)
89 \( 1 + (268.5 + 465. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (446 - 772. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.07829863158123358199235413281, −9.632061930947750656436341572558, −8.905891401573734055014846366794, −7.999868453354922183931629732675, −6.40293416707224975160559555569, −5.71359397433451210606528334101, −4.65219524793157897230766611049, −3.04598135996937827272461945861, −2.38026550489331008153648201260, −1.01713968434453822484568122939, 1.28217631351688089116720078464, 2.01123501734342055054080408903, 3.64450512169865314240829137527, 5.38733462904612682929608695598, 5.96605270237494583686473589707, 6.72959360174637360251911509134, 7.74827720145902738869162626763, 8.783972077000982732390476669093, 9.514794734678228818953473809976, 10.19346896092073684303120209908

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