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Properties

Label 2-546-273.38-c1-0-22
Degree $2$
Conductor $546$
Sign $0.836 + 0.547i$
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 + (1.40 + 1.00i)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 + (0.962 + 2.84i)9-s + (−2.19 − 1.26i)10-s + (1.32 − 2.29i)11-s + (−1.57 + 0.714i)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 + (0.812 + 0.582i)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 + (0.320 + 0.947i)9-s + (−0.694 − 0.401i)10-s + (0.399 − 0.691i)11-s + (−0.455 + 0.206i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.836 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68806 - 0.503287i\)
\(L(\frac12)\) \(\approx\) \(1.68806 - 0.503287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 1.00i)T \)
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

−10.54244511482750033482097834580, −9.673847780644381011237352033614, −9.073948698479619548688514401933, −8.629744904522540606172346706793, −7.27633480366374476205504793554, −6.04267138360889540974201575516, −4.93195997951652574774431637217, −3.60225041698895726265122825878, −2.86744114337097503200411539903, −1.32096779119976915930334224526, 1.53370963165962760349091248822, 2.81912694556838706471596669945, 4.03854266327602179309050929501, 6.00957290807407113236862906798, 6.26495011938757957023518214629, 7.24166563817733496734180520280, 8.161880222797723886424943098639, 9.274415617833964480168286933808, 9.631255539410474123526797027006, 10.49910997085160439545283979464

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