Properties

Label 2-546-91.80-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.894 + 0.446i$
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 + (−1.13 + 4.23i)5-s + (−0.965 + 0.258i)6-s + (0.0297 − 2.64i)7-s + (0.707 + 0.707i)8-s − 9-s + 4.38·10-s + (−2.28 − 2.28i)11-s + (0.499 + 0.866i)12-s + (2.90 − 2.13i)13-s + (−2.56 + 0.656i)14-s + (4.23 + 1.13i)15-s + (0.500 − 0.866i)16-s + (−3.17 − 5.49i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (−0.507 + 1.89i)5-s + (−0.394 + 0.105i)6-s + (0.0112 − 0.999i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 1.38·10-s + (−0.688 − 0.688i)11-s + (0.144 + 0.249i)12-s + (0.804 − 0.593i)13-s + (−0.685 + 0.175i)14-s + (1.09 + 0.292i)15-s + (0.125 − 0.216i)16-s + (−0.769 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.894 + 0.446i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157285 - 0.666945i\)
\(L(\frac12)\) \(\approx\) \(0.157285 - 0.666945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0297 + 2.64i)T \)
13 \( 1 + (-2.90 + 2.13i)T \)
good5 \( 1 + (1.13 - 4.23i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.28 + 2.28i)T + 11iT^{2} \)
17 \( 1 + (3.17 + 5.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.82 + 3.82i)T + 19iT^{2} \)
23 \( 1 + (-1.71 - 0.992i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.25 + 2.21i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.173 - 0.0465i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.00297 + 0.0111i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.69 + 1.55i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.71 - 0.994i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.22 - 1.66i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 5.70iT - 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 + (-1.61 - 6.00i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.931 - 3.47i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.17 - 1.17i)T + 83iT^{2} \)
89 \( 1 + (4.61 + 17.2i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.25 - 0.603i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.76489190697979143226550687665, −9.914393854244971465957978202289, −8.511237016193105888934926507360, −7.62977535367178974053541601810, −6.98097004269514367981555622859, −6.09961015665845727628612398938, −4.37162980839672343290378177802, −3.18339577558919218647998194596, −2.56863267948297093434973462780, −0.41802059967969967065394941468, 1.74226951065525582950203578777, 3.96684031183156784710555682813, 4.70498645551366543852113825710, 5.49686044497782780286644786893, 6.45241695536133686208077254508, 8.176009011372308735918127103989, 8.425011050533433846554060799494, 9.096062162460552618702480340087, 10.01079040891903013522421199131, 11.15272608548581755838785829980

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