Properties

Label 2-546-91.33-c1-0-16
Degree $2$
Conductor $546$
Sign $-0.721 - 0.692i$
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  + (0.258 − 0.965i)2-s + i·3-s + (−0.866 − 0.499i)4-s + (−0.620 − 2.31i)5-s + (0.965 + 0.258i)6-s + (−2.53 − 0.751i)7-s + (−0.707 + 0.707i)8-s − 9-s − 2.39·10-s + (−2.37 + 2.37i)11-s + (0.499 − 0.866i)12-s + (−2.14 + 2.89i)13-s + (−1.38 + 2.25i)14-s + (2.31 − 0.620i)15-s + (0.500 + 0.866i)16-s + (−3.18 + 5.51i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (−0.277 − 1.03i)5-s + (0.394 + 0.105i)6-s + (−0.958 − 0.284i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 0.757·10-s + (−0.716 + 0.716i)11-s + (0.144 − 0.249i)12-s + (−0.595 + 0.803i)13-s + (−0.369 + 0.602i)14-s + (0.597 − 0.160i)15-s + (0.125 + 0.216i)16-s + (−0.772 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.721 - 0.692i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00480329 + 0.0119446i\)
\(L(\frac12)\) \(\approx\) \(0.00480329 + 0.0119446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 - iT \)
7 \( 1 + (2.53 + 0.751i)T \)
13 \( 1 + (2.14 - 2.89i)T \)
good5 \( 1 + (0.620 + 2.31i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.37 - 2.37i)T - 11iT^{2} \)
17 \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.51 + 4.51i)T - 19iT^{2} \)
23 \( 1 + (5.63 - 3.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.87 + 3.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.84 + 1.56i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.05 - 0.818i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.36 + 8.82i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.35 - 4.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.00 - 2.14i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.65 + 8.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.37 - 0.635i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 7.54iT - 61T^{2} \)
67 \( 1 + (-5.28 - 5.28i)T + 67iT^{2} \)
71 \( 1 + (-0.165 + 0.616i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.86 + 10.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.75 - 4.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.287 - 0.287i)T - 83iT^{2} \)
89 \( 1 + (-0.715 + 2.66i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.55 - 0.953i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.05947790311265773547650937214, −9.611245438029725115590538521538, −8.801395441422752871427679563926, −7.72826800976682885591393329366, −6.45761706731132641742126157645, −5.17030774470051583725813892321, −4.44375000264754518346030155810, −3.53263297633366702337301727205, −2.05308278953979140466011828660, −0.00652316323078885407942829904, 2.75340978063206851174449483763, 3.36581609400042536605893714541, 5.13474263463993739033738525980, 6.04267667675938397708045649627, 6.86795435805389796241227173009, 7.56755066856545126172091550911, 8.391890768644387443306513013905, 9.600130100574844482550987342866, 10.38222997714830110143880800656, 11.44289428560277727785834156258

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