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Properties

Label 2-546-273.173-c1-0-37
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: $\chi_{546} (173, \cdot )$
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

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

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