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Properties

Label 2-546-91.19-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.993 - 0.116i$
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  + (0.965 + 0.258i)2-s − i·3-s + (0.866 + 0.499i)4-s + (−3.15 + 0.844i)5-s + (0.258 − 0.965i)6-s + (−2.64 − 0.0462i)7-s + (0.707 + 0.707i)8-s − 9-s − 3.26·10-s + (−0.954 − 0.954i)11-s + (0.499 − 0.866i)12-s + (−3.41 + 1.16i)13-s + (−2.54 − 0.729i)14-s + (0.844 + 3.15i)15-s + (0.500 + 0.866i)16-s + (−1.17 + 2.03i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s − 0.577i·3-s + (0.433 + 0.249i)4-s + (−1.40 + 0.377i)5-s + (0.105 − 0.394i)6-s + (−0.999 − 0.0174i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s − 1.03·10-s + (−0.287 − 0.287i)11-s + (0.144 − 0.249i)12-s + (−0.946 + 0.322i)13-s + (−0.679 − 0.194i)14-s + (0.218 + 0.813i)15-s + (0.125 + 0.216i)16-s + (−0.285 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00464016 + 0.0796563i\)
\(L(\frac12)\) \(\approx\) \(0.00464016 + 0.0796563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + iT \)
7 \( 1 + (2.64 + 0.0462i)T \)
13 \( 1 + (3.41 - 1.16i)T \)
good5 \( 1 + (3.15 - 0.844i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.954 + 0.954i)T + 11iT^{2} \)
17 \( 1 + (1.17 - 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.26 + 3.26i)T + 19iT^{2} \)
23 \( 1 + (-0.786 + 0.453i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.41 - 7.64i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.496 + 1.85i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.95 + 11.0i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.56 - 0.687i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.72 - 0.998i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.88 - 7.02i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.56 - 6.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.55 + 5.81i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.730iT - 61T^{2} \)
67 \( 1 + (-8.73 + 8.73i)T - 67iT^{2} \)
71 \( 1 + (2.14 + 0.576i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.21 + 1.66i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.18 - 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.48 + 2.48i)T + 83iT^{2} \)
89 \( 1 + (4.45 + 1.19i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.33 - 8.72i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.24658886104954350730180114332, −10.79521794425005701577416817009, −9.346073375048363641186654432045, −8.309245852914147132862851021241, −7.28719376113384622046404582070, −6.92269401839316285651493240296, −5.81961421225749618117285648108, −4.48031968727429431835230669823, −3.52666012524459127279149743588, −2.52939592298579819672363083159, 0.03406102587664645520789043343, 2.67275132911820060010292383542, 3.74300019041510237735586451137, 4.45706742836045879103743979216, 5.45157842605625221300305075876, 6.70273431973726995625284354121, 7.63750110137376808597362110314, 8.559512069090474441794071360953, 9.776375621639750767872116803637, 10.32939422157083129991770023241

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