Properties

Label 2-546-21.20-c1-0-18
Degree $2$
Conductor $546$
Sign $-0.487 + 0.872i$
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 + (−0.618 + 1.61i)3-s − 4-s − 3.23·5-s + (1.61 + 0.618i)6-s + (2.61 + 0.381i)7-s + i·8-s + (−2.23 − 2.00i)9-s + 3.23i·10-s + (0.618 − 1.61i)12-s i·13-s + (0.381 − 2.61i)14-s + (2.00 − 5.23i)15-s + 16-s − 2.47·17-s + (−2.00 + 2.23i)18-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.356 + 0.934i)3-s − 0.5·4-s − 1.44·5-s + (0.660 + 0.252i)6-s + (0.989 + 0.144i)7-s + 0.353i·8-s + (−0.745 − 0.666i)9-s + 1.02i·10-s + (0.178 − 0.467i)12-s − 0.277i·13-s + (0.102 − 0.699i)14-s + (0.516 − 1.35i)15-s + 0.250·16-s − 0.599·17-s + (−0.471 + 0.527i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.487 + 0.872i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.487 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283132 - 0.482645i\)
\(L(\frac12)\) \(\approx\) \(0.283132 - 0.482645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.61i)T \)
7 \( 1 + (-2.61 - 0.381i)T \)
13 \( 1 + iT \)
good5 \( 1 + 3.23T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 + 7.23iT - 31T^{2} \)
37 \( 1 + 9.23T + 37T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 + 3.23iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 4.47iT - 61T^{2} \)
67 \( 1 - 6.94T + 67T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 - 3.23iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 - 0.763iT - 97T^{2} \)
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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

−10.82920534452968617411785385247, −9.807778948701095473042969965002, −8.693734720871506082321274230757, −8.216534980517118743093663847432, −6.98742680793422053909581370811, −5.43111265036818444640130116022, −4.49143929834660022589351200282, −3.96840321484916764839688827600, −2.63288593148908599925937862563, −0.35882198135387177808338923728, 1.50523886031027302027082048232, 3.54211697031364059506847117997, 4.67832086198914608700981358427, 5.59447061058866085654829214665, 6.91491668874169121820254979907, 7.40443081121605419777750268475, 8.257794059892876196197614131795, 8.719472560662486399619692099993, 10.47019959921805842096265426944, 11.26441287951958042426053598679

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