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Properties

Label 2-546-21.17-c1-0-22
Degree $2$
Conductor $546$
Sign $0.823 + 0.566i$
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  + (0.866 − 0.5i)2-s + (1.70 + 0.301i)3-s + (0.499 − 0.866i)4-s + (−0.737 − 1.27i)5-s + (1.62 − 0.591i)6-s + (2.24 + 1.39i)7-s − 0.999i·8-s + (2.81 + 1.02i)9-s + (−1.27 − 0.737i)10-s + (−0.616 − 0.356i)11-s + (1.11 − 1.32i)12-s − i·13-s + (2.64 + 0.0805i)14-s + (−0.872 − 2.40i)15-s + (−0.5 − 0.866i)16-s + (−0.629 + 1.09i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.984 + 0.174i)3-s + (0.249 − 0.433i)4-s + (−0.329 − 0.571i)5-s + (0.664 − 0.241i)6-s + (0.850 + 0.526i)7-s − 0.353i·8-s + (0.939 + 0.343i)9-s + (−0.404 − 0.233i)10-s + (−0.186 − 0.107i)11-s + (0.321 − 0.382i)12-s − 0.277i·13-s + (0.706 + 0.0215i)14-s + (−0.225 − 0.620i)15-s + (−0.125 − 0.216i)16-s + (−0.152 + 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69838 - 0.838213i\)
\(L(\frac12)\) \(\approx\) \(2.69838 - 0.838213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-1.70 - 0.301i)T \)
7 \( 1 + (-2.24 - 1.39i)T \)
13 \( 1 + iT \)
good5 \( 1 + (0.737 + 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.616 + 0.356i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.629 - 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.83 - 1.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.87 - 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.97iT - 29T^{2} \)
31 \( 1 + (1.42 + 0.824i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.738 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.74T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 + (1.02 + 1.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.4 - 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.61 - 2.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.50 - 6.07i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 + (1.31 + 0.760i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + (1.87 + 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.98iT - 97T^{2} \)
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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

−10.70599604946586700967957155061, −9.917843500714634640656047556383, −8.728436244679493470367239844079, −8.302549474557895978334043757445, −7.28459503905079236296188752984, −5.85044491634696106391927510133, −4.78354433425766725728779870963, −4.04699538681733053547632104771, −2.78208156327261532859764058776, −1.63913993037126078315295808376, 1.90004435150792238988729326551, 3.17507222947525591012931070823, 4.12822643389629442132228474919, 5.07069988648856417830475590596, 6.65378939267138223214746744953, 7.24769936562846423998241570600, 8.063232397880029928265953648339, 8.828044189077739422479746651388, 10.05956318627922962457948755818, 10.96591342641091591217073696447

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