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Properties

Label 2-546-273.230-c1-0-28
Degree $2$
Conductor $546$
Sign $-0.176 + 0.984i$
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  + (0.866 + 0.5i)2-s + (−0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s − 3.46·5-s + (−1.5 + 0.866i)6-s + (0.5 − 2.59i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (−2.99 − 1.73i)10-s − 1.73·12-s + (−3.5 − 0.866i)13-s + (1.73 − 2i)14-s + (2.99 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (−0.866 − 1.5i)17-s − 3i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s − 1.54·5-s + (−0.612 + 0.353i)6-s + (0.188 − 0.981i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.948 − 0.547i)10-s − 0.499·12-s + (−0.970 − 0.240i)13-s + (0.462 − 0.534i)14-s + (0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (−0.210 − 0.363i)17-s − 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.176 + 0.984i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177262 - 0.211848i\)
\(L(\frac12)\) \(\approx\) \(0.177262 - 0.211848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (3.5 + 0.866i)T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.19 + 3i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.46 + 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.9 - 7.5i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (-7.79 + 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15 + 8.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75139170633053949212889825840, −9.883465942565673539831727228393, −8.629744702930262615261391829276, −7.60344448091282754521744260281, −7.07999264952234701067234775681, −5.73958790636977049398061183689, −4.51926108991097106991413323029, −4.19562865801136918239219619837, −3.12689028378477764426356222157, −0.13325000368667494612849200963, 1.91026716998056003792026557517, 3.19850324291206216120454854724, 4.50867574293029526605818923561, 5.37595688880259021863038615697, 6.46644902166198543807753762332, 7.45826450457202090943083417192, 8.101375645850671921809143863692, 9.202879693435580770945586793884, 10.62396293736644550176822939130, 11.49309761553062599419825577907

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