Properties

Label 2-546-273.101-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.613 + 0.789i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.801 + 1.53i)3-s + (−0.499 − 0.866i)4-s + (−1.09 + 0.634i)5-s + (−1.73 − 0.0741i)6-s + (−2.21 + 1.45i)7-s + 0.999·8-s + (−1.71 + 2.46i)9-s − 1.26i·10-s − 5.15·11-s + (0.929 − 1.46i)12-s + (2.55 − 2.53i)13-s + (−0.151 − 2.64i)14-s + (−1.85 − 1.17i)15-s + (−0.5 + 0.866i)16-s + (−2.50 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.462 + 0.886i)3-s + (−0.249 − 0.433i)4-s + (−0.491 + 0.283i)5-s + (−0.706 − 0.0302i)6-s + (−0.835 + 0.548i)7-s + 0.353·8-s + (−0.572 + 0.820i)9-s − 0.401i·10-s − 1.55·11-s + (0.268 − 0.421i)12-s + (0.709 − 0.704i)13-s + (−0.0404 − 0.705i)14-s + (−0.478 − 0.304i)15-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169960 - 0.347393i\)
\(L(\frac12)\) \(\approx\) \(0.169960 - 0.347393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.801 - 1.53i)T \)
7 \( 1 + (2.21 - 1.45i)T \)
13 \( 1 + (-2.55 + 2.53i)T \)
good5 \( 1 + (1.09 - 0.634i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.15T + 11T^{2} \)
17 \( 1 + (2.50 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.60T + 19T^{2} \)
23 \( 1 + (2.33 + 1.35i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.776 - 0.448i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.25 - 7.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.79 + 2.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.54 - 2.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.72 + 2.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.18 - 2.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.7 - 6.76i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.2 - 5.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.66iT - 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 + (-3.79 + 6.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.210 - 0.365i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.95 + 5.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.22iT - 83T^{2} \)
89 \( 1 + (-7.19 - 4.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.21 - 10.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98054312363243128087248726878, −10.36131011407355101657157607221, −9.497173961747498196737161752087, −8.766688830887147877856361215311, −7.86227060985377683154751622192, −7.11071010371698150844580000603, −5.62631156145745610148519071949, −5.11956824199126060458318886012, −3.54529452385534240525798932206, −2.74404335501813459191616220137, 0.22486748046216150538100880839, 1.87462238786083771899180930383, 3.18696510427599820409561108117, 4.04862962700493943996673312525, 5.69728425551016949696590988475, 6.85561033689179997112490432297, 7.76614208483826080819077635451, 8.337649347649139157509601076564, 9.357637178681420719645824341912, 10.19653876962594004059396746296

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