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Properties

Label 2-546-273.242-c1-0-34
Degree $2$
Conductor $546$
Sign $-0.605 - 0.796i$
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  + (−0.965 − 0.258i)2-s + (1.15 − 1.29i)3-s + (0.866 + 0.499i)4-s + (−2.59 − 0.696i)5-s + (−1.44 + 0.948i)6-s + (−2.64 − 0.123i)7-s + (−0.707 − 0.707i)8-s + (−0.336 − 2.98i)9-s + (2.33 + 1.34i)10-s + (−2.18 + 0.584i)11-s + (1.64 − 0.541i)12-s + (3.60 + 0.177i)13-s + (2.52 + 0.803i)14-s + (−3.89 + 2.55i)15-s + (0.500 + 0.866i)16-s + (−2.76 + 4.78i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.666 − 0.745i)3-s + (0.433 + 0.249i)4-s + (−1.16 − 0.311i)5-s + (−0.591 + 0.387i)6-s + (−0.998 − 0.0467i)7-s + (−0.249 − 0.249i)8-s + (−0.112 − 0.993i)9-s + (0.736 + 0.425i)10-s + (−0.658 + 0.176i)11-s + (0.474 − 0.156i)12-s + (0.998 + 0.0491i)13-s + (0.673 + 0.214i)14-s + (−1.00 + 0.659i)15-s + (0.125 + 0.216i)16-s + (−0.670 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.605 - 0.796i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.605 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00683043 + 0.0137729i\)
\(L(\frac12)\) \(\approx\) \(0.00683043 + 0.0137729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.15 + 1.29i)T \)
7 \( 1 + (2.64 + 0.123i)T \)
13 \( 1 + (-3.60 - 0.177i)T \)
good5 \( 1 + (2.59 + 0.696i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.18 - 0.584i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 - 4.24i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.278 + 0.482i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.74iT - 29T^{2} \)
31 \( 1 + (6.07 - 1.62i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (5.40 + 1.44i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.93 + 2.93i)T - 41iT^{2} \)
43 \( 1 - 4.32iT - 43T^{2} \)
47 \( 1 + (-2.24 + 8.37i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.23 + 5.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.68 + 0.720i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.48 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.6 - 3.38i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (11.0 - 11.0i)T - 71iT^{2} \)
73 \( 1 + (-0.788 - 2.94i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.11 - 5.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (0.622 - 2.32i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.99 + 1.99i)T + 97iT^{2} \)
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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

−10.23058930114909574060468331616, −9.018756385305623214330530331169, −8.471763494915299090507336887394, −7.74498967194013372356533469195, −6.87743214840964822144087910584, −5.96260415016808666428712638396, −3.94791222198606670922588597349, −3.31050670855477626323217838665, −1.75969054456531008711479349160, −0.009736610923491427907540366554, 2.66427776996780329005907501436, 3.49220049489421803774378281930, 4.61760519377290019427063238186, 6.03287956327443901923863190026, 7.23377357431709581479309519781, 7.83628058771214657864804713751, 8.944033477532222260147192655712, 9.308236655823271842553930794044, 10.56448380933622979882651760938, 10.98510192887880982435099468284

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