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Properties

Label 2-546-91.19-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.435 - 0.900i$
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 + (−1.43 + 0.384i)5-s + (0.258 − 0.965i)6-s + (1.74 + 1.98i)7-s + (−0.707 − 0.707i)8-s − 9-s + 1.48·10-s + (−0.00182 − 0.00182i)11-s + (−0.499 + 0.866i)12-s + (3.56 − 0.558i)13-s + (−1.17 − 2.37i)14-s + (−0.384 − 1.43i)15-s + (0.500 + 0.866i)16-s + (−3.27 + 5.67i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (−0.642 + 0.172i)5-s + (0.105 − 0.394i)6-s + (0.660 + 0.750i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.469·10-s + (−0.000549 − 0.000549i)11-s + (−0.144 + 0.249i)12-s + (0.987 − 0.155i)13-s + (−0.314 − 0.633i)14-s + (−0.0993 − 0.370i)15-s + (0.125 + 0.216i)16-s + (−0.795 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.435 - 0.900i$
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: \((2,\ 546,\ (\ :1/2),\ -0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416516 + 0.664206i\)
\(L(\frac12)\) \(\approx\) \(0.416516 + 0.664206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 - 1.98i)T \)
13 \( 1 + (-3.56 + 0.558i)T \)
good5 \( 1 + (1.43 - 0.384i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.00182 + 0.00182i)T + 11iT^{2} \)
17 \( 1 + (3.27 - 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.18 + 1.18i)T + 19iT^{2} \)
23 \( 1 + (-0.438 + 0.253i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.810 - 1.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.18 - 8.17i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.20 + 4.48i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.86 - 1.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.25 - 3.03i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.507 + 1.89i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.92 - 6.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.944 + 3.52i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 + (-2.46 + 2.46i)T - 67iT^{2} \)
71 \( 1 + (11.1 + 2.98i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.87 - 1.57i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.83 + 4.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.65 - 8.65i)T + 83iT^{2} \)
89 \( 1 + (-12.9 - 3.46i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.02 + 15.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.93314718278460970385508033802, −10.43772077223269988159100458773, −9.071114505600682408101853694113, −8.606939294952525700209876438876, −7.85491467652462948593089790082, −6.60418057216775462519520807244, −5.57389859084917536846500087775, −4.28720490098407178906811453556, −3.27808113161091907527295309695, −1.78176932611502771169947583778, 0.57145406022236949178831061801, 2.02145289625852308784455499582, 3.71630571898083084754036582627, 4.87914326037572667336384546970, 6.23201076178233411860712503278, 7.12279365729052996182631185264, 7.88560270715282601115536233794, 8.509952731704690644332007163391, 9.511865614921090134437121288276, 10.61303761670057669120129477238

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