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Properties

Label 2-546-273.44-c1-0-17
Degree $2$
Conductor $546$
Sign $0.872 - 0.487i$
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 + (0.541 + 1.64i)3-s + (0.866 − 0.499i)4-s + (2.59 − 0.696i)5-s + (0.948 + 1.44i)6-s + (−2.64 + 0.123i)7-s + (0.707 − 0.707i)8-s + (−2.41 + 1.78i)9-s + (2.33 − 1.34i)10-s + (2.18 + 0.584i)11-s + (1.29 + 1.15i)12-s + (3.60 − 0.177i)13-s + (−2.52 + 0.803i)14-s + (2.55 + 3.89i)15-s + (0.500 − 0.866i)16-s + (2.76 + 4.78i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.312 + 0.949i)3-s + (0.433 − 0.249i)4-s + (1.16 − 0.311i)5-s + (0.387 + 0.591i)6-s + (−0.998 + 0.0467i)7-s + (0.249 − 0.249i)8-s + (−0.804 + 0.593i)9-s + (0.736 − 0.425i)10-s + (0.658 + 0.176i)11-s + (0.372 + 0.333i)12-s + (0.998 − 0.0491i)13-s + (−0.673 + 0.214i)14-s + (0.659 + 1.00i)15-s + (0.125 − 0.216i)16-s + (0.670 + 1.16i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.872 - 0.487i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.872 - 0.487i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57618 + 0.670984i\)
\(L(\frac12)\) \(\approx\) \(2.57618 + 0.670984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.541 - 1.64i)T \)
7 \( 1 + (2.64 - 0.123i)T \)
13 \( 1 + (-3.60 + 0.177i)T \)
good5 \( 1 + (-2.59 + 0.696i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.18 - 0.584i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.76 - 4.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.13 + 4.24i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.278 + 0.482i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.74iT - 29T^{2} \)
31 \( 1 + (6.07 + 1.62i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.40 - 1.44i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.93 + 2.93i)T + 41iT^{2} \)
43 \( 1 + 4.32iT - 43T^{2} \)
47 \( 1 + (2.24 + 8.37i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.23 + 5.32i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.68 + 0.720i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.48 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.6 + 3.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.0 - 11.0i)T + 71iT^{2} \)
73 \( 1 + (-0.788 + 2.94i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.11 + 5.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.5 + 11.5i)T + 83iT^{2} \)
89 \( 1 + (-0.622 - 2.32i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.99 - 1.99i)T - 97iT^{2} \)
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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

−10.57213210978932589436848763101, −10.17586586958356468866852078258, −9.157395689093946621350724005634, −8.686231476418549136024087104530, −6.92557832375621596865111105299, −5.94887380296348757613259524563, −5.36457208053434527887214765179, −4.01634104774602222388017426828, −3.26731256366636813899368770541, −1.86447607368029530212142152445, 1.52892160159114470193350478035, 2.83523621126846240069855316500, 3.70582787370571829718422619945, 5.58178194584407566406737042064, 6.19002915249598104594074767412, 6.78673099806159017613260143652, 7.81865708496329120684478882016, 9.038424021311875265945406474288, 9.711904227131753132912384239790, 10.83632810947424126033272420559

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