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Properties

Label 2-546-91.89-c1-0-15
Degree $2$
Conductor $546$
Sign $0.678 + 0.735i$
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  + (0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s + 1.00i·4-s + (−1.08 − 4.03i)5-s + (0.258 + 0.965i)6-s + (−1.31 − 2.29i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (2.09 − 3.62i)10-s + (−0.163 − 0.611i)11-s + (−0.500 + 0.866i)12-s + (−1.96 − 3.02i)13-s + (0.697 − 2.55i)14-s + (1.08 − 4.03i)15-s − 1.00·16-s + 5.91·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s + 0.500i·4-s + (−0.483 − 1.80i)5-s + (0.105 + 0.394i)6-s + (−0.495 − 0.868i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.660 − 1.14i)10-s + (−0.0494 − 0.184i)11-s + (−0.144 + 0.250i)12-s + (−0.544 − 0.838i)13-s + (0.186 − 0.682i)14-s + (0.279 − 1.04i)15-s − 0.250·16-s + 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.678 + 0.735i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.678 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64582 - 0.720945i\)
\(L(\frac12)\) \(\approx\) \(1.64582 - 0.720945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.31 + 2.29i)T \)
13 \( 1 + (1.96 + 3.02i)T \)
good5 \( 1 + (1.08 + 4.03i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.163 + 0.611i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + (-2.38 - 0.639i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.14iT - 23T^{2} \)
29 \( 1 + (0.692 + 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.35 - 1.43i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.59 - 1.59i)T - 37iT^{2} \)
41 \( 1 + (1.22 + 0.327i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.40 + 1.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.20 + 0.859i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.96 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.53 - 8.53i)T + 59iT^{2} \)
61 \( 1 + (10.7 - 6.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.13 - 1.64i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-14.8 + 3.98i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.14 + 8.01i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.213 - 0.370i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.72 + 5.72i)T - 83iT^{2} \)
89 \( 1 + (1.18 + 1.18i)T + 89iT^{2} \)
97 \( 1 + (1.95 + 7.30i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.46951544282122841924437892102, −9.698753010461426330572286901434, −8.714983435132145544588667680739, −7.973619107041924174179534560396, −7.36094355082063873495506913156, −5.80988969077598169076360297871, −4.93166840479291337919793679843, −4.14276666733988603704538266650, −3.14127828437144604083245041693, −0.868254229817543035919780385402, 2.15011746090222023228642218692, 3.06875811531412825214800132782, 3.71490060376538417340273559930, 5.37634638657939027328037679627, 6.48412526753694132935283016140, 7.16651431994299091733236108003, 8.086160714656388085547822970029, 9.595887610830226445339394492925, 9.916618908785446716541558487267, 11.11807106085217891372364807870

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