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Properties

Label 2-546-91.24-c1-0-13
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: $\chi_{546} (115, \cdot )$
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

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

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