Properties

Label 2-546-273.272-c1-0-35
Degree $2$
Conductor $546$
Sign $-0.0179 + 0.999i$
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  + 2-s + (0.420 − 1.68i)3-s + 4-s − 3.36i·5-s + (0.420 − 1.68i)6-s + (2.37 + 1.16i)7-s + 8-s + (−2.64 − 1.41i)9-s − 3.36i·10-s + (0.420 − 1.68i)12-s + (−2.79 + 2.27i)13-s + (2.37 + 1.16i)14-s + (−5.64 − 1.41i)15-s + 16-s + 7.82·17-s + (−2.64 − 1.41i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.242 − 0.970i)3-s + 0.5·4-s − 1.50i·5-s + (0.171 − 0.685i)6-s + (0.898 + 0.439i)7-s + 0.353·8-s + (−0.881 − 0.471i)9-s − 1.06i·10-s + (0.121 − 0.485i)12-s + (−0.775 + 0.631i)13-s + (0.635 + 0.311i)14-s + (−1.45 − 0.365i)15-s + 0.250·16-s + 1.89·17-s + (−0.623 − 0.333i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0179 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0179 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0179 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0179 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72468 - 1.75591i\)
\(L(\frac12)\) \(\approx\) \(1.72468 - 1.75591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.420 + 1.68i)T \)
7 \( 1 + (-2.37 - 1.16i)T \)
13 \( 1 + (2.79 - 2.27i)T \)
good5 \( 1 + 3.36iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 + 5.59T + 19T^{2} \)
23 \( 1 - 0.500iT - 23T^{2} \)
29 \( 1 - 5.15iT - 29T^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 - 2.32iT - 37T^{2} \)
41 \( 1 + 9.87iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 4.33iT - 47T^{2} \)
53 \( 1 - 0.500iT - 53T^{2} \)
59 \( 1 - 2.16iT - 59T^{2} \)
61 \( 1 + 4.55iT - 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 0.708T + 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 - 3.14iT - 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−10.88622034364657183804022532616, −9.445352933014817954611494375851, −8.589843072029457245480849805136, −7.933946874840838441613128566635, −7.00229672427072776411756769395, −5.62957038901126388328901625048, −5.15554748690114524894287604568, −3.95464164930227558061773156770, −2.29242286878425419671947571755, −1.26678522265415791953678982884, 2.39596921077032745336579671841, 3.32161872110317103064426462375, 4.26993994497284867154739850293, 5.32630506671858817082303792858, 6.27111190366325200432948894749, 7.57031332206715839726420363776, 8.010689368999383434167241600007, 9.646033464796044553947535224953, 10.41022646194682906230407080332, 10.86594669906221360717023471263

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