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Properties

Label 2-546-91.51-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.00340 - 0.999i$
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 + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (2.18 + 1.26i)5-s + 0.999i·6-s + (−1.47 + 2.19i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.26 + 2.18i)10-s + (4.88 − 2.82i)11-s + (−0.499 + 0.866i)12-s + (−3.13 − 1.78i)13-s + (−2.37 + 1.16i)14-s + 2.52i·15-s + (−0.5 + 0.866i)16-s + (0.123 + 0.214i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (0.976 + 0.563i)5-s + 0.408i·6-s + (−0.556 + 0.830i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.398 + 0.690i)10-s + (1.47 − 0.850i)11-s + (−0.144 + 0.249i)12-s + (−0.869 − 0.493i)13-s + (−0.634 + 0.311i)14-s + 0.650i·15-s + (−0.125 + 0.216i)16-s + (0.0300 + 0.0519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00340 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.00340 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.00340 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74890 + 1.75485i\)
\(L(\frac12)\) \(\approx\) \(1.74890 + 1.75485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.47 - 2.19i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (-2.18 - 1.26i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.88 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.123 - 0.214i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 + 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.80 - 3.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + (-5.30 + 3.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.01 - 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 5.56T + 43T^{2} \)
47 \( 1 + (-5.78 - 3.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.28 + 5.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.89 - 3.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.39 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.09i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.06iT - 71T^{2} \)
73 \( 1 + (6.81 - 3.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.22 + 5.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.662iT - 83T^{2} \)
89 \( 1 + (4.86 + 2.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.8iT - 97T^{2} \)
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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

−11.07972189883646827075881000447, −9.948693717772826610000946051085, −9.345339518427348518734174360789, −8.503620500053025437944908098945, −7.20087504938312730242152138616, −6.05490136742601297177733154946, −5.82556099407861533955238706017, −4.36073782053483567572511849246, −3.20364629258551445349692156894, −2.29359339384885112555973728525, 1.31977872919570090902761125662, 2.36557320225474341248318481163, 3.92280660484104105408865221362, 4.71196049769353842008465122454, 6.16259155833391021639882890649, 6.67403787533775443359364412697, 7.71451274434396492128328034774, 9.292328647427683371462303066606, 9.523749801424637218379790527894, 10.51463986244091829742242876983

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