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Properties

Label 2-546-91.74-c1-0-7
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: $\chi_{546} (529, \cdot )$
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

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

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