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Properties

Label 2-546-91.4-c1-0-5
Degree $2$
Conductor $546$
Sign $0.759 - 0.649i$
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  − i·2-s + (−0.5 − 0.866i)3-s − 4-s + (−1.54 + 0.889i)5-s + (−0.866 + 0.5i)6-s + (−2.51 − 0.824i)7-s + i·8-s + (−0.499 + 0.866i)9-s + (0.889 + 1.54i)10-s + (−0.0253 + 0.0146i)11-s + (0.5 + 0.866i)12-s + (0.100 + 3.60i)13-s + (−0.824 + 2.51i)14-s + (1.54 + 0.889i)15-s + 16-s + 6.11·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (−0.689 + 0.397i)5-s + (−0.353 + 0.204i)6-s + (−0.950 − 0.311i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.281 + 0.487i)10-s + (−0.00763 + 0.00440i)11-s + (0.144 + 0.249i)12-s + (0.0279 + 0.999i)13-s + (−0.220 + 0.671i)14-s + (0.397 + 0.229i)15-s + 0.250·16-s + 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.759 - 0.649i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.759 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583241 + 0.215387i\)
\(L(\frac12)\) \(\approx\) \(0.583241 + 0.215387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.51 + 0.824i)T \)
13 \( 1 + (-0.100 - 3.60i)T \)
good5 \( 1 + (1.54 - 0.889i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0253 - 0.0146i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 6.11T + 17T^{2} \)
19 \( 1 + (-5.76 - 3.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.80T + 23T^{2} \)
29 \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.81 + 2.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.9iT - 37T^{2} \)
41 \( 1 + (6.40 + 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.27 - 7.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.84 - 2.79i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0633 + 0.109i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.7iT - 59T^{2} \)
61 \( 1 + (-3.21 + 5.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.93 - 2.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.24 - 4.18i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.21 - 3.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.798 - 1.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.39iT - 83T^{2} \)
89 \( 1 - 13.5iT - 89T^{2} \)
97 \( 1 + (9.75 - 5.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.08149152463237133183574573885, −9.975132736791863262941085205725, −9.517500239279163001650781346152, −8.078633289717845684050626046118, −7.39021364729706126792577368637, −6.39614098057812489908614087290, −5.30705972475024432016824429716, −3.81616961096189214323675937680, −3.17587687316644701151607882252, −1.44800263051182617950403741514, 0.39998145223160067208283961722, 3.16452470102695014192850167108, 4.02499374663371815128793531189, 5.40114419679391273006034447383, 5.81881002284223407941415897747, 7.21168464439538212024783146070, 7.912942983578285439248078240438, 8.942390504088199322538092111060, 9.760309983289239155716042793137, 10.43730876514231825859838946569

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