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Properties

Label 2-546-91.33-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.227 - 0.973i$
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  + (0.258 − 0.965i)2-s + i·3-s + (−0.866 − 0.499i)4-s + (0.133 + 0.499i)5-s + (0.965 + 0.258i)6-s + (−0.932 + 2.47i)7-s + (−0.707 + 0.707i)8-s − 9-s + 0.516·10-s + (−3.33 + 3.33i)11-s + (0.499 − 0.866i)12-s + (−1.50 − 3.27i)13-s + (2.15 + 1.54i)14-s + (−0.499 + 0.133i)15-s + (0.500 + 0.866i)16-s + (0.668 − 1.15i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.0598 + 0.223i)5-s + (0.394 + 0.105i)6-s + (−0.352 + 0.935i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s + 0.163·10-s + (−1.00 + 1.00i)11-s + (0.144 − 0.249i)12-s + (−0.416 − 0.909i)13-s + (0.574 + 0.411i)14-s + (−0.128 + 0.0345i)15-s + (0.125 + 0.216i)16-s + (0.162 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.227 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.515720 + 0.650335i\)
\(L(\frac12)\) \(\approx\) \(0.515720 + 0.650335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 - iT \)
7 \( 1 + (0.932 - 2.47i)T \)
13 \( 1 + (1.50 + 3.27i)T \)
good5 \( 1 + (-0.133 - 0.499i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.33 - 3.33i)T - 11iT^{2} \)
17 \( 1 + (-0.668 + 1.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.91 - 5.91i)T - 19iT^{2} \)
23 \( 1 + (3.54 - 2.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.91 - 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.93 - 0.785i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.03 - 0.546i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.92 - 7.18i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.73 + 5.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.25 - 1.67i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.49 - 4.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.00 - 0.270i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + (-3.58 - 3.58i)T + 67iT^{2} \)
71 \( 1 + (-1.85 + 6.93i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.804 + 3.00i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.47 - 3.47i)T - 83iT^{2} \)
89 \( 1 + (-1.80 + 6.71i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-14.7 - 3.94i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.80249371869854943028504507286, −10.22950792206800822914018770565, −9.635128343850103759494968647589, −8.554423809690201834625781063729, −7.68661339997998931422941420335, −6.16858177653199580028295806704, −5.34120227168910949338951597820, −4.38628288756509785662241689267, −3.06698892577205582725799072972, −2.21696753289147145699933677557, 0.42603478527046812084699968264, 2.51244141757175069202055321574, 3.96822134106862171333569449374, 4.98220901668027001776915900837, 6.19348882536336562126425893447, 6.83576115774446360253845790530, 7.78364413684850361589981966285, 8.536141947142980124592000880817, 9.489552923300835004772147609842, 10.64482668605175240107133827732

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