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Properties

Label 2-546-91.41-c1-0-12
Degree $2$
Conductor $546$
Sign $0.887 - 0.460i$
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 + (0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (2.50 + 2.50i)5-s + (0.965 − 0.258i)6-s + (−1.87 − 1.86i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.77 + 3.06i)10-s + (−0.883 + 3.29i)11-s + 12-s + (1.87 − 3.08i)13-s + (−1.32 − 2.28i)14-s + (3.41 + 0.916i)15-s + (0.500 + 0.866i)16-s + (−0.906 + 1.57i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (1.11 + 1.11i)5-s + (0.394 − 0.105i)6-s + (−0.709 − 0.705i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.559 + 0.969i)10-s + (−0.266 + 0.993i)11-s + 0.288·12-s + (0.518 − 0.854i)13-s + (−0.355 − 0.611i)14-s + (0.882 + 0.236i)15-s + (0.125 + 0.216i)16-s + (−0.219 + 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.887 - 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72322 + 0.664759i\)
\(L(\frac12)\) \(\approx\) \(2.72322 + 0.664759i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (1.87 + 1.86i)T \)
13 \( 1 + (-1.87 + 3.08i)T \)
good5 \( 1 + (-2.50 - 2.50i)T + 5iT^{2} \)
11 \( 1 + (0.883 - 3.29i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.906 - 1.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.44 + 1.72i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.86 - 3.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.24 + 3.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.69 + 6.69i)T + 31iT^{2} \)
37 \( 1 + (-0.544 + 2.03i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.43 - 5.35i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.23 + 1.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.09 + 2.09i)T - 47iT^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + (-0.202 - 0.753i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.49 + 2.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.6 + 3.13i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.20 + 4.48i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-4.59 + 4.59i)T - 73iT^{2} \)
79 \( 1 - 8.32T + 79T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + (6.87 + 1.84i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.49 + 0.401i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.71925853176903160000809746875, −9.997346290600912569211371597114, −9.418557374564260978183682141890, −7.70128592579457937486466467114, −7.25669084453866307288231424364, −6.24917261075135170315303712148, −5.56904774413443644412325455171, −3.92264709062449569988600305156, −3.02777960833541359769789408250, −1.96379759232745674623012129776, 1.59379829682392479623482979330, 2.83897002556809528450419081902, 3.96023697865023473115282282827, 5.35628654830092438590245660845, 5.69127632321221425224722948128, 6.85826877138938558780540679562, 8.423744143211347630125075785127, 9.074087375768042296945535548290, 9.698067008246533692061644106251, 10.66117070810307229771494944844

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