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Properties

Label 2-546-39.32-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.813 + 0.582i$
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 + (−0.211 − 1.71i)3-s + (0.866 − 0.499i)4-s + (0.0940 + 0.0940i)5-s + (0.649 + 1.60i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (−2.91 + 0.726i)9-s + (−0.115 − 0.0664i)10-s + (−0.597 − 2.23i)11-s + (−1.04 − 1.38i)12-s + (1.75 − 3.14i)13-s − i·14-s + (0.141 − 0.181i)15-s + (0.500 − 0.866i)16-s + (−0.998 − 1.72i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.122 − 0.992i)3-s + (0.433 − 0.249i)4-s + (0.0420 + 0.0420i)5-s + (0.264 + 0.655i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.970 + 0.242i)9-s + (−0.0364 − 0.0210i)10-s + (−0.180 − 0.672i)11-s + (−0.300 − 0.399i)12-s + (0.487 − 0.872i)13-s − 0.267i·14-s + (0.0366 − 0.0468i)15-s + (0.125 − 0.216i)16-s + (−0.242 − 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183495 - 0.571457i\)
\(L(\frac12)\) \(\approx\) \(0.183495 - 0.571457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.211 + 1.71i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-1.75 + 3.14i)T \)
good5 \( 1 + (-0.0940 - 0.0940i)T + 5iT^{2} \)
11 \( 1 + (0.597 + 2.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.998 + 1.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.63 + 0.975i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.925 + 1.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.64 + 2.68i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.47 - 3.47i)T - 31iT^{2} \)
37 \( 1 + (5.91 - 1.58i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.86 + 1.57i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.57 - 3.79i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.38 - 1.38i)T - 47iT^{2} \)
53 \( 1 + 9.54iT - 53T^{2} \)
59 \( 1 + (3.63 + 0.974i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.39 + 7.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.191 - 0.714i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.76 + 10.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.50 - 4.50i)T + 73iT^{2} \)
79 \( 1 - 9.50T + 79T^{2} \)
83 \( 1 + (-4.90 - 4.90i)T + 83iT^{2} \)
89 \( 1 + (-2.38 - 8.91i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.719 - 0.192i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.64578117846763980979909908596, −9.358512887716892104842179992212, −8.437531247983581226798939873759, −7.970793183743942283111615677569, −6.79128653866647412158137066815, −6.13465572011373746782876199483, −5.18228486002680090756815709367, −3.21189422470901169210036291213, −2.04964071918910538017832574860, −0.41951408427438723131837238632, 1.89162980412077023305330818368, 3.50077732885068777375657601360, 4.35431111641447134906929167708, 5.60606481520973518479571608160, 6.70564520792686327751406722722, 7.71095089331122081624629767140, 8.913367674781799341518573861464, 9.307549461597385872512790095998, 10.33474262287836451798482955712, 10.89405580497305176596282839752

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