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Properties

Label 2-546-273.38-c1-0-27
Degree $2$
Conductor $546$
Sign $0.660 + 0.750i$
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 + (−1.31 − 1.13i)3-s + (−0.499 + 0.866i)4-s + (1.08 − 0.627i)5-s + (0.326 − 1.70i)6-s + (1.14 − 2.38i)7-s − 0.999·8-s + (0.432 + 2.96i)9-s + (1.08 + 0.627i)10-s + (−0.900 + 1.56i)11-s + (1.63 − 0.568i)12-s + (−0.657 − 3.54i)13-s + (2.63 − 0.204i)14-s + (−2.13 − 0.409i)15-s + (−0.5 − 0.866i)16-s + (1.12 − 1.95i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.756 − 0.654i)3-s + (−0.249 + 0.433i)4-s + (0.486 − 0.280i)5-s + (0.133 − 0.694i)6-s + (0.431 − 0.902i)7-s − 0.353·8-s + (0.144 + 0.989i)9-s + (0.343 + 0.198i)10-s + (−0.271 + 0.470i)11-s + (0.472 − 0.164i)12-s + (−0.182 − 0.983i)13-s + (0.704 − 0.0545i)14-s + (−0.551 − 0.105i)15-s + (−0.125 − 0.216i)16-s + (0.273 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.660 + 0.750i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.660 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23195 - 0.556932i\)
\(L(\frac12)\) \(\approx\) \(1.23195 - 0.556932i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 1.13i)T \)
7 \( 1 + (-1.14 + 2.38i)T \)
13 \( 1 + (0.657 + 3.54i)T \)
good5 \( 1 + (-1.08 + 0.627i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.900 - 1.56i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.12 + 1.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.46 + 4.26i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.93 + 4.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.62iT - 29T^{2} \)
31 \( 1 + (-2.39 + 4.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.77 - 2.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.52iT - 41T^{2} \)
43 \( 1 - 3.14T + 43T^{2} \)
47 \( 1 + (-8.20 + 4.73i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.2 - 6.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.80 + 2.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.4 + 6.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.21T + 71T^{2} \)
73 \( 1 + (7.95 - 13.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.323 + 0.560i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + (-11.1 + 6.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.48T + 97T^{2} \)
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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

−10.74722900925990318360759305772, −9.951721934140575731466980004768, −8.666970509405201587015107258868, −7.61493906316129785375276857424, −7.11542656336546683854484578313, −6.08388367259455386391762638579, −5.11291448615058020121038317214, −4.48435681020459073595690013340, −2.60330907778161076094974393176, −0.814030117408671583112464598708, 1.68430431240835531386615989220, 3.12086064032729403717962892042, 4.30533334238521434909720216028, 5.40317346189444405507196175869, 5.89021707269638724534620375106, 7.05004758642213421071939676971, 8.756583752530527931126793105639, 9.214334153614561924329557075432, 10.50083490127244028411219917798, 10.69915198566393046406473177631

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