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Properties

Label 2-546-273.32-c1-0-19
Degree $2$
Conductor $546$
Sign $0.977 - 0.209i$
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  + (0.707 + 0.707i)2-s + (−1.31 − 1.12i)3-s + 1.00i·4-s + (−0.258 + 0.965i)5-s + (−0.136 − 1.72i)6-s + (−0.443 − 2.60i)7-s + (−0.707 + 0.707i)8-s + (0.470 + 2.96i)9-s + (−0.865 + 0.499i)10-s + (2.62 + 0.704i)11-s + (1.12 − 1.31i)12-s + (3.59 − 0.238i)13-s + (1.53 − 2.15i)14-s + (1.42 − 0.981i)15-s − 1.00·16-s + 2.39·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.760 − 0.649i)3-s + 0.500i·4-s + (−0.115 + 0.431i)5-s + (−0.0556 − 0.704i)6-s + (−0.167 − 0.985i)7-s + (−0.250 + 0.250i)8-s + (0.156 + 0.987i)9-s + (−0.273 + 0.158i)10-s + (0.792 + 0.212i)11-s + (0.324 − 0.380i)12-s + (0.997 − 0.0661i)13-s + (0.409 − 0.576i)14-s + (0.368 − 0.253i)15-s − 0.250·16-s + 0.580·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53258 + 0.162628i\)
\(L(\frac12)\) \(\approx\) \(1.53258 + 0.162628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.31 + 1.12i)T \)
7 \( 1 + (0.443 + 2.60i)T \)
13 \( 1 + (-3.59 + 0.238i)T \)
good5 \( 1 + (0.258 - 0.965i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.62 - 0.704i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 + (0.807 + 3.01i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.02T + 23T^{2} \)
29 \( 1 + (-2.49 - 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.354 + 1.32i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.770 - 0.770i)T + 37iT^{2} \)
41 \( 1 + (0.406 - 0.108i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.05 - 1.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.717 - 0.192i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.12 + 1.22i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.69 + 7.69i)T - 59iT^{2} \)
61 \( 1 + (0.903 - 1.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.116 + 0.0311i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.68 - 13.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.19 + 1.12i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.50 + 7.50i)T + 83iT^{2} \)
89 \( 1 + (7.22 - 7.22i)T - 89iT^{2} \)
97 \( 1 + (4.73 + 1.26i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.09431780288134724827516530351, −10.20756596884361812418357826154, −8.877980658958725355776197760400, −7.78091528303281187978395671761, −6.85047086992523896965555478906, −6.58603469121550726114407344082, −5.35108565638502260562236576879, −4.30489948826815058111260679864, −3.14676926020871760523626231772, −1.15698919045244015206561708210, 1.20765146743833754914619842389, 3.11812918366163257804806090225, 4.09954996106298038004896461572, 5.14325439131355427827006718107, 5.91987641560020379524668465897, 6.69848357193673925502836543581, 8.518179504159250987128376274664, 9.109274573728827934750940142599, 10.04254742169690323410291244554, 10.95515601943021057152329533948

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