Properties

Label 2-546-273.173-c1-0-31
Degree $2$
Conductor $546$
Sign $-0.467 + 0.884i$
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  + (−0.5 − 0.866i)2-s + (1.70 − 0.329i)3-s + (−0.499 + 0.866i)4-s + (−1.26 − 0.730i)5-s + (−1.13 − 1.30i)6-s + (−2.63 − 0.263i)7-s + 0.999·8-s + (2.78 − 1.11i)9-s + 1.46i·10-s + 3.51·11-s + (−0.565 + 1.63i)12-s + (−0.214 − 3.59i)13-s + (1.08 + 2.41i)14-s + (−2.39 − 0.825i)15-s + (−0.5 − 0.866i)16-s + (3.79 − 6.57i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.981 − 0.190i)3-s + (−0.249 + 0.433i)4-s + (−0.565 − 0.326i)5-s + (−0.463 − 0.533i)6-s + (−0.995 − 0.0995i)7-s + 0.353·8-s + (0.927 − 0.373i)9-s + 0.461i·10-s + 1.06·11-s + (−0.163 + 0.472i)12-s + (−0.0595 − 0.998i)13-s + (0.290 + 0.644i)14-s + (−0.617 − 0.213i)15-s + (−0.125 − 0.216i)16-s + (0.920 − 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.467 + 0.884i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.467 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.669483 - 1.11122i\)
\(L(\frac12)\) \(\approx\) \(0.669483 - 1.11122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.70 + 0.329i)T \)
7 \( 1 + (2.63 + 0.263i)T \)
13 \( 1 + (0.214 + 3.59i)T \)
good5 \( 1 + (1.26 + 0.730i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 3.51T + 11T^{2} \)
17 \( 1 + (-3.79 + 6.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.45T + 19T^{2} \)
23 \( 1 + (3.12 - 1.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.170 - 0.0985i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.34 + 9.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.81 - 2.78i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.60 - 1.50i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.61 - 9.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.0 - 6.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.21 + 2.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.13 + 0.654i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.999iT - 61T^{2} \)
67 \( 1 - 5.52iT - 67T^{2} \)
71 \( 1 + (5.33 + 9.24i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.94 - 5.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.174 + 0.302i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.72iT - 83T^{2} \)
89 \( 1 + (12.5 - 7.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.72 - 2.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.22123037024391381956473755857, −9.546889679239655623024297364500, −8.962938109984424534775972667659, −7.87370255936422288471411552072, −7.31899222009736015817335211831, −6.02470659951332169380147862112, −4.30454373410917988764590270018, −3.52243644944560581164674051798, −2.54408036750560515193510124453, −0.78511934367260255449300457027, 1.86934505383141657897356166543, 3.64047348794899939715630440637, 4.03988775251221073196738233820, 5.82864825936161459681358393325, 6.84492802268775936996690599092, 7.41250746459370275005597773352, 8.758026803318974056471521968279, 8.912667209102204624937500778087, 10.08553459302989797148922516763, 10.67843564003152987938367660977

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