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Properties

Label 2-546-273.62-c1-0-35
Degree $2$
Conductor $546$
Sign $-0.998 + 0.0514i$
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 + (−1.17 − 1.27i)3-s + (−0.499 + 0.866i)4-s − 3.58i·5-s + (0.514 − 1.65i)6-s + (−2.54 − 0.727i)7-s − 0.999·8-s + (−0.237 + 2.99i)9-s + (3.10 − 1.79i)10-s + (0.630 + 1.09i)11-s + (1.68 − 0.381i)12-s + (−2.88 + 2.16i)13-s + (−0.641 − 2.56i)14-s + (−4.55 + 4.21i)15-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.73i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.678 − 0.734i)3-s + (−0.249 + 0.433i)4-s − 1.60i·5-s + (0.209 − 0.675i)6-s + (−0.961 − 0.275i)7-s − 0.353·8-s + (−0.0791 + 0.996i)9-s + (0.981 − 0.566i)10-s + (0.190 + 0.329i)11-s + (0.487 − 0.110i)12-s + (−0.799 + 0.600i)13-s + (−0.171 − 0.686i)14-s + (−1.17 + 1.08i)15-s + (−0.125 − 0.216i)16-s + (−0.383 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0514i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.998 + 0.0514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00551360 - 0.214133i\)
\(L(\frac12)\) \(\approx\) \(0.00551360 - 0.214133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 1.27i)T \)
7 \( 1 + (2.54 + 0.727i)T \)
13 \( 1 + (2.88 - 2.16i)T \)
good5 \( 1 + 3.58iT - 5T^{2} \)
11 \( 1 + (-0.630 - 1.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.57 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.386 + 0.670i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.28 - 4.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.76 + 3.90i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + (0.424 - 0.244i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.17 - 0.676i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.125 + 0.216i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + (10.4 + 6.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.27 - 1.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0759 + 0.131i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 + (-5.69 + 3.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.21 - 3.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.21269565345890876175965735275, −9.369498872111427169435439275419, −8.423345768493863026444591646798, −7.52869037403425892124360261429, −6.62021693932721619114736994537, −5.77292569222319799537932123233, −4.87231156776481420801745032039, −3.97475289156810681115963015190, −1.87516244740067605025888295854, −0.11291620809100415554961459964, 2.68836744386692392489452564631, 3.31865437139977920702251493269, 4.47684782542132087300213172740, 5.82280398930132261769646629721, 6.38001793433019799810856359982, 7.36535601413886445918087703185, 9.034846292887155566618854399047, 9.891965155697786582657809755113, 10.47777297943691989012939386336, 11.00584958354264409506797384544

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