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Properties

Label 2-546-91.47-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.671 + 0.740i$
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 + (−0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.930 − 3.47i)5-s + (−0.707 + 0.707i)6-s + (−2.03 + 1.68i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.79 − 3.11i)10-s + (0.653 + 0.175i)11-s + (−0.5 + 0.866i)12-s + (−2.65 − 2.44i)13-s + (−1.53 + 2.15i)14-s + (2.54 + 2.54i)15-s + (0.500 − 0.866i)16-s + (−2.74 − 4.76i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.416 − 1.55i)5-s + (−0.288 + 0.288i)6-s + (−0.770 + 0.636i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.568 − 0.984i)10-s + (0.197 + 0.0528i)11-s + (−0.144 + 0.249i)12-s + (−0.735 − 0.677i)13-s + (−0.409 + 0.576i)14-s + (0.656 + 0.656i)15-s + (0.125 − 0.216i)16-s + (−0.666 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438080 - 0.988616i\)
\(L(\frac12)\) \(\approx\) \(0.438080 - 0.988616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.03 - 1.68i)T \)
13 \( 1 + (2.65 + 2.44i)T \)
good5 \( 1 + (0.930 + 3.47i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.653 - 0.175i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.74 + 4.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00922 - 0.0344i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.63 + 3.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 + (2.56 + 0.688i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.558 + 2.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.23 + 3.23i)T - 41iT^{2} \)
43 \( 1 - 6.92iT - 43T^{2} \)
47 \( 1 + (-8.99 + 2.41i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.982 + 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.52 - 5.70i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.80 - 2.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.39 + 5.22i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-9.32 - 9.32i)T + 71iT^{2} \)
73 \( 1 + (-0.906 + 3.38i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.22 + 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.11 + 1.11i)T - 83iT^{2} \)
89 \( 1 + (-5.80 + 1.55i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (11.0 - 11.0i)T - 97iT^{2} \)
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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

−10.53688551889192161967870150307, −9.567402020338749443331083027398, −8.919283150920128197457478123175, −7.78862799344899603523154868632, −6.54653898941928239075695433121, −5.52819058234504417180505193277, −4.85308490000536817449826959070, −3.98391785192409487863192960551, −2.50793978573718737051388067625, −0.50566229200358196701972653953, 2.26936931316293762011582761000, 3.53601228172812947308009532152, 4.30136543520650360123949532843, 5.89392281186151725763421133808, 6.66661668990253870461590723147, 7.08829612220756789117820779297, 8.051766977092733140887263743703, 9.680784081432375329014223178661, 10.54083074267299133409238575746, 11.10827463722179258445912577729

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