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Properties

Label 2-546-13.3-c1-0-0
Degree $2$
Conductor $546$
Sign $0.794 - 0.607i$
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 3.56·5-s + (−0.499 + 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.78 + 3.08i)10-s + (−1.28 − 2.21i)11-s + 0.999·12-s + (−0.5 + 3.57i)13-s − 0.999·14-s + (1.78 + 3.08i)15-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s − 1.59·5-s + (−0.204 + 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.563 + 0.975i)10-s + (−0.386 − 0.668i)11-s + 0.288·12-s + (−0.138 + 0.990i)13-s − 0.267·14-s + (0.459 + 0.796i)15-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428275 + 0.144947i\)
\(L(\frac12)\) \(\approx\) \(0.428275 + 0.144947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 3.57i)T \)
good5 \( 1 + 3.56T + 5T^{2} \)
11 \( 1 + (1.28 + 2.21i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.84 - 6.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.56 - 7.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + (-4.90 - 8.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.06 + 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.43 + 2.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 + 3.87T + 53T^{2} \)
59 \( 1 + (6.56 - 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.561 + 0.972i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2 + 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (-4.84 - 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.12 - 10.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15673592121477558475301328700, −10.22402275174936494403593107152, −9.057893418106304410325546294972, −7.999741098589306161321626467936, −7.66105301476937064089661115836, −6.60013991705357099876328372533, −5.08134273399618757730208809428, −4.00787513167301698174705292594, −3.05429821191537438574459869072, −1.23128666803714783207566773468, 0.35327922923447364433279661753, 2.91628802504928215060906728393, 4.36404261140080671322979389177, 4.88948270922628189036482240660, 6.23718076521617461231642562774, 7.23336688417032258737000325406, 8.152169742345082598544790999765, 8.606913685448065839120454845725, 9.849873184753851268962566376937, 10.79584529436428909261531621424

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