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Properties

Label 2-546-273.152-c1-0-25
Degree $2$
Conductor $546$
Sign $-0.0206 + 0.999i$
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 + (1.30 − 1.13i)3-s + (0.499 − 0.866i)4-s + (−0.604 + 1.04i)5-s + (−0.562 + 1.63i)6-s + (−2.45 − 0.985i)7-s + 0.999i·8-s + (0.412 − 2.97i)9-s − 1.20i·10-s + 1.19i·11-s + (−0.331 − 1.69i)12-s + (−1.92 − 3.04i)13-s + (2.61 − 0.374i)14-s + (0.401 + 2.05i)15-s + (−0.5 − 0.866i)16-s + (2.61 − 4.53i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.754 − 0.656i)3-s + (0.249 − 0.433i)4-s + (−0.270 + 0.468i)5-s + (−0.229 + 0.668i)6-s + (−0.928 − 0.372i)7-s + 0.353i·8-s + (0.137 − 0.990i)9-s − 0.382i·10-s + 0.361i·11-s + (−0.0958 − 0.490i)12-s + (−0.533 − 0.845i)13-s + (0.699 − 0.100i)14-s + (0.103 + 0.530i)15-s + (−0.125 − 0.216i)16-s + (0.635 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0206 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0206 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0206 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0206 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677928 - 0.692104i\)
\(L(\frac12)\) \(\approx\) \(0.677928 - 0.692104i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 1.13i)T \)
7 \( 1 + (2.45 + 0.985i)T \)
13 \( 1 + (1.92 + 3.04i)T \)
good5 \( 1 + (0.604 - 1.04i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.19iT - 11T^{2} \)
17 \( 1 + (-2.61 + 4.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.35iT - 19T^{2} \)
23 \( 1 + (-3.13 + 1.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.43 + 0.827i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.45 - 1.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.50 + 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0914 + 0.158i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.527 - 0.913i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.56 - 6.17i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.91 + 1.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.63 - 9.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 6.61iT - 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 + (6.08 - 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.40 + 3.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.23 + 3.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 + (3.59 + 6.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.8 + 6.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.38754194467492696200742337140, −9.414489790659368887533210929671, −8.965598060273053271172339630264, −7.48405585490288351420464053179, −7.33129273252376350374402922086, −6.47118205498306207148950522687, −5.07997948292981884379777840548, −3.36492798317901169784576234616, −2.57860418914950197244159630658, −0.61640258229694017705502023852, 1.84360315471094486988623576525, 3.25813767359402298515077530023, 3.97139715663789347100646190993, 5.36490106324374961837533421149, 6.62493846331432634667454194703, 7.88192612954215447958627052616, 8.512150034843526274059795652979, 9.332443865207458054260498757185, 9.974890833147013083480726226800, 10.69711076080143769248743026376

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