Properties

Label 2-546-91.16-c1-0-17
Degree $2$
Conductor $546$
Sign $0.172 + 0.984i$
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  + 2-s + (0.5 − 0.866i)3-s + 4-s + (2.05 − 3.56i)5-s + (0.5 − 0.866i)6-s + (−2.61 − 0.405i)7-s + 8-s + (−0.499 − 0.866i)9-s + (2.05 − 3.56i)10-s + (−2.02 + 3.50i)11-s + (0.5 − 0.866i)12-s + (1.81 − 3.11i)13-s + (−2.61 − 0.405i)14-s + (−2.05 − 3.56i)15-s + 16-s + 0.715·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (0.920 − 1.59i)5-s + (0.204 − 0.353i)6-s + (−0.988 − 0.153i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.650 − 1.12i)10-s + (−0.609 + 1.05i)11-s + (0.144 − 0.249i)12-s + (0.503 − 0.864i)13-s + (−0.698 − 0.108i)14-s + (−0.531 − 0.920i)15-s + 0.250·16-s + 0.173·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89907 - 1.59473i\)
\(L(\frac12)\) \(\approx\) \(1.89907 - 1.59473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.61 + 0.405i)T \)
13 \( 1 + (-1.81 + 3.11i)T \)
good5 \( 1 + (-2.05 + 3.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.02 - 3.50i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.715T + 17T^{2} \)
19 \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.09T + 23T^{2} \)
29 \( 1 + (-4.50 - 7.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.82 - 3.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.19T + 37T^{2} \)
41 \( 1 + (2.88 + 4.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.28 + 2.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.28 + 2.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.35 + 2.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + (-4.71 - 8.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.583 + 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.10 + 8.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.25 + 2.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.29496003809635143199674653530, −9.877149256652699618909460988969, −8.784382641997591303656331507651, −7.949035810322898926068670179728, −6.81143068436600290075299991102, −5.76650197971405561473276077051, −5.16636489253378511224028201203, −3.90404185854603470782363961804, −2.52873782507111515886733830617, −1.22026042075076759750175896581, 2.50219283918103932188771879971, 3.02494871357376634597234412654, 4.10909715035425819767296493703, 5.76040371384825407668634480706, 6.19812679093395992140498683117, 7.05339114202378628113182970841, 8.315752331400220450911219040977, 9.741539892178935748990397724069, 9.956696303530243781130531344809, 11.12577172881725641688037841081

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