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Properties

Label 2-546-13.10-c3-0-10
Degree $2$
Conductor $546$
Sign $-0.972 - 0.234i$
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 + 17.5i·5-s + (5.19 + 3i)6-s + (6.06 + 3.5i)7-s + 7.99i·8-s + (−4.5 + 7.79i)9-s + (−17.5 − 30.4i)10-s + (−6.24 + 3.60i)11-s − 12·12-s + (34.3 + 31.8i)13-s − 14·14-s + (45.6 − 26.3i)15-s + (−8 − 13.8i)16-s + (−51.0 + 88.4i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 1.57i·5-s + (0.353 + 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.555 − 0.962i)10-s + (−0.171 + 0.0988i)11-s − 0.288·12-s + (0.733 + 0.679i)13-s − 0.267·14-s + (0.785 − 0.453i)15-s + (−0.125 − 0.216i)16-s + (−0.728 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.972 - 0.234i$
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: \((2,\ 546,\ (\ :3/2),\ -0.972 - 0.234i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8942171239\)
\(L(\frac12)\) \(\approx\) \(0.8942171239\)
\(L(\frac{5}{2})\) 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 \( 1 + (-34.3 - 31.8i)T \)
good5 \( 1 - 17.5iT - 125T^{2} \)
11 \( 1 + (6.24 - 3.60i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (51.0 - 88.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-50.3 - 29.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.10 + 5.38i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-40.4 - 70.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 64.8iT - 2.97e4T^{2} \)
37 \( 1 + (-326. + 188. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (70.9 - 40.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-186. + 322. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 26.6iT - 1.03e5T^{2} \)
53 \( 1 + 573.T + 1.48e5T^{2} \)
59 \( 1 + (47.3 + 27.3i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (427. - 740. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (288. - 166. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (59.1 + 34.1i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 367. iT - 3.89e5T^{2} \)
79 \( 1 + 348.T + 4.93e5T^{2} \)
83 \( 1 - 573. iT - 5.71e5T^{2} \)
89 \( 1 + (478. - 276. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.06e3 + 613. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−10.94569229813021984449878629529, −10.05141346773340988502321165740, −8.928453174998758634820360567413, −7.956362272868843660307272708613, −7.17747048961217891656624702028, −6.38745391930495772007883212902, −5.77033278529554854105778568131, −4.07830180075148359186979786764, −2.65344870306081477652622026390, −1.55802145025687545062476320606, 0.37693315576185041678476958823, 1.28108438006986518610589739076, 2.99476784715980967478837230245, 4.43471079090055380635318406537, 5.04080493762906169671767354719, 6.20297088112267538656192774574, 7.66392258485828280971441151892, 8.391672059872499443832812191823, 9.230518158838631500426374597520, 9.761167575698489477189886423332

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