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Properties

Label 2-546-13.4-c3-0-36
Degree $2$
Conductor $546$
Sign $0.267 + 0.963i$
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 + 4.88i·5-s + (5.19 − 3i)6-s + (6.06 − 3.5i)7-s + 7.99i·8-s + (−4.5 − 7.79i)9-s + (−4.88 + 8.45i)10-s + (−36.6 − 21.1i)11-s + 12·12-s + (−33.0 − 33.2i)13-s + 14·14-s + (12.6 + 7.32i)15-s + (−8 + 13.8i)16-s + (−39.4 − 68.4i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + 0.436i·5-s + (0.353 − 0.204i)6-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.154 + 0.267i)10-s + (−1.00 − 0.579i)11-s + 0.288·12-s + (−0.704 − 0.709i)13-s + 0.267·14-s + (0.218 + 0.126i)15-s + (−0.125 + 0.216i)16-s + (−0.563 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.267 + 0.963i$
Analytic conductor: \(32.2150\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :3/2),\ 0.267 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.403835198\)
\(L(\frac12)\) \(\approx\) \(2.403835198\)
\(L(\frac{5}{2})\) 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 \( 1 + (33.0 + 33.2i)T \)
good5 \( 1 - 4.88iT - 125T^{2} \)
11 \( 1 + (36.6 + 21.1i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (39.4 + 68.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-78.5 + 45.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-78.6 + 136. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-20.6 + 35.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 83.7iT - 2.97e4T^{2} \)
37 \( 1 + (333. + 192. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-200. - 115. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-185. - 321. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 511. iT - 1.03e5T^{2} \)
53 \( 1 + 222.T + 1.48e5T^{2} \)
59 \( 1 + (-18.0 + 10.4i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-214. - 371. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-213. - 123. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-305. + 176. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 + 827.T + 4.93e5T^{2} \)
83 \( 1 + 385. iT - 5.71e5T^{2} \)
89 \( 1 + (-503. - 290. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (651. - 376. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−10.45750753370626385915376907006, −9.131368524572776924986169593817, −8.183436897312230569019321398587, −7.33747281760036307519186909817, −6.76129473614876688952566783770, −5.44736618866656842283396678472, −4.75056071829865287307084110977, −3.10801073681823969207315134247, −2.54373280460927705819768484754, −0.54586186047156486281973773826, 1.57754234419680108906924536264, 2.71593435948590895508087390993, 3.95294260253997954837987568391, 4.92947326629886195206442532454, 5.50142003369473183743458949760, 6.97309809256976760153474902700, 7.920086633682233043013575716782, 8.996601211879406679148133930093, 9.766943738296669689608052888353, 10.62461912517818169022591125678

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