Properties

Label 2-546-91.19-c1-0-9
Degree $2$
Conductor $546$
Sign $0.497 - 0.867i$
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 + i·3-s + (0.866 + 0.499i)4-s + (0.421 − 0.112i)5-s + (−0.258 + 0.965i)6-s + (2.44 + 1.01i)7-s + (0.707 + 0.707i)8-s − 9-s + 0.435·10-s + (1.13 + 1.13i)11-s + (−0.499 + 0.866i)12-s + (−1.06 − 3.44i)13-s + (2.09 + 1.61i)14-s + (0.112 + 0.421i)15-s + (0.500 + 0.866i)16-s + (0.582 − 1.00i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (0.188 − 0.0504i)5-s + (−0.105 + 0.394i)6-s + (0.923 + 0.382i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.137·10-s + (0.342 + 0.342i)11-s + (−0.144 + 0.249i)12-s + (−0.294 − 0.955i)13-s + (0.561 + 0.430i)14-s + (0.0291 + 0.108i)15-s + (0.125 + 0.216i)16-s + (0.141 − 0.244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07816 + 1.20414i\)
\(L(\frac12)\) \(\approx\) \(2.07816 + 1.20414i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-2.44 - 1.01i)T \)
13 \( 1 + (1.06 + 3.44i)T \)
good5 \( 1 + (-0.421 + 0.112i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.13 - 1.13i)T + 11iT^{2} \)
17 \( 1 + (-0.582 + 1.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.57 - 4.57i)T + 19iT^{2} \)
23 \( 1 + (4.85 - 2.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.92 - 3.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.28 + 8.51i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.27 + 4.75i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.78 + 0.478i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.251 - 0.145i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.19 - 4.45i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.94 + 5.10i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.35 + 12.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.413iT - 61T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - 67iT^{2} \)
71 \( 1 + (7.98 + 2.13i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.797 - 0.213i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.79 + 8.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.03 + 7.03i)T + 83iT^{2} \)
89 \( 1 + (-6.66 - 1.78i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.433 + 1.61i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.14093433013424887204935835360, −10.02898914530384930222712458697, −9.361401360751074720836770977749, −7.994792161870790042923975808084, −7.56675404668787516065781750895, −5.89694607394990459556172057317, −5.44365386249805867127789308131, −4.37538520849658770849994626672, −3.33716928961606332642437662324, −1.89920729236626309922963002973, 1.35789824949306518013654124536, 2.55687199109741235795247897658, 4.02183346117751847937539681853, 4.94521487689308050671381599736, 6.05348143721031117987126153669, 6.93880079865209409691114658961, 7.80047396609241968897981892961, 8.805521954254993574839963290248, 9.931320940827915293938312815155, 10.90490189844393929658667135820

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