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Properties

Label 2-546-273.194-c1-0-20
Degree $2$
Conductor $546$
Sign $0.878 - 0.477i$
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  + (−0.5 + 0.866i)2-s + (1.57 − 0.714i)3-s + (−0.499 − 0.866i)4-s + (2.19 + 1.26i)5-s + (−0.170 + 1.72i)6-s + (2.61 − 0.373i)7-s + 0.999·8-s + (1.97 − 2.25i)9-s + (−2.19 + 1.26i)10-s + (1.32 + 2.29i)11-s + (−1.40 − 1.00i)12-s + (−3.59 + 0.207i)13-s + (−0.986 + 2.45i)14-s + (4.37 + 0.432i)15-s + (−0.5 + 0.866i)16-s + (−1.84 − 3.19i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.911 − 0.412i)3-s + (−0.249 − 0.433i)4-s + (0.982 + 0.567i)5-s + (−0.0695 + 0.703i)6-s + (0.989 − 0.141i)7-s + 0.353·8-s + (0.659 − 0.751i)9-s + (−0.694 + 0.401i)10-s + (0.399 + 0.691i)11-s + (−0.406 − 0.291i)12-s + (−0.998 + 0.0575i)13-s + (−0.263 + 0.656i)14-s + (1.12 + 0.111i)15-s + (−0.125 + 0.216i)16-s + (−0.447 − 0.775i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.878 - 0.477i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.878 - 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95090 + 0.495554i\)
\(L(\frac12)\) \(\approx\) \(1.95090 + 0.495554i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.57 + 0.714i)T \)
7 \( 1 + (-2.61 + 0.373i)T \)
13 \( 1 + (3.59 - 0.207i)T \)
good5 \( 1 + (-2.19 - 1.26i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.84 + 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.241 + 0.418i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.09 + 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.93iT - 29T^{2} \)
31 \( 1 + (-2.71 - 4.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.90 + 3.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.03iT - 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 + (6.66 + 3.84i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.13 + 4.69i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.98 - 2.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.25 - 2.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.38 + 0.797i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-6.57 - 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.75 - 6.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.10iT - 83T^{2} \)
89 \( 1 + (2.37 + 1.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.91T + 97T^{2} \)
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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

−10.46445469165280995257164816144, −9.846063643381908534466702756181, −9.024772757335845809658800258468, −8.196214685991172341147224571195, −7.09438988291975441117455383893, −6.82311555513964896564351611853, −5.37992791382173483356300635527, −4.33968936078015920793521613775, −2.58564920704474824632117891687, −1.65961586092263684235149832183, 1.62001072643893403329386159587, 2.43978063881124021999846302889, 3.92965393214413707649937510006, 4.86152414852208740514808768771, 5.95665808298642026200990893287, 7.61242603903211954409427520318, 8.319757827463370784015951929966, 9.082494615794564027764295610162, 9.768006457474523371226759621343, 10.48681711137794038874262454923

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