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Properties

Label 2-546-273.242-c1-0-24
Degree $2$
Conductor $546$
Sign $0.909 - 0.416i$
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  + (0.965 + 0.258i)2-s + (1.70 + 0.306i)3-s + (0.866 + 0.499i)4-s + (−1.82 − 0.488i)5-s + (1.56 + 0.737i)6-s + (2.57 − 0.619i)7-s + (0.707 + 0.707i)8-s + (2.81 + 1.04i)9-s + (−1.63 − 0.944i)10-s + (−1.71 + 0.458i)11-s + (1.32 + 1.11i)12-s + (1.62 + 3.22i)13-s + (2.64 + 0.0671i)14-s + (−2.96 − 1.39i)15-s + (0.500 + 0.866i)16-s + (2.01 − 3.48i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.984 + 0.177i)3-s + (0.433 + 0.249i)4-s + (−0.816 − 0.218i)5-s + (0.639 + 0.301i)6-s + (0.972 − 0.234i)7-s + (0.249 + 0.249i)8-s + (0.937 + 0.348i)9-s + (−0.517 − 0.298i)10-s + (−0.516 + 0.138i)11-s + (0.381 + 0.322i)12-s + (0.449 + 0.893i)13-s + (0.706 + 0.0179i)14-s + (−0.764 − 0.359i)15-s + (0.125 + 0.216i)16-s + (0.488 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77836 + 0.605941i\)
\(L(\frac12)\) \(\approx\) \(2.77836 + 0.605941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.70 - 0.306i)T \)
7 \( 1 + (-2.57 + 0.619i)T \)
13 \( 1 + (-1.62 - 3.22i)T \)
good5 \( 1 + (1.82 + 0.488i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.71 - 0.458i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.01 + 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.854 - 3.18i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.75 + 6.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.36iT - 29T^{2} \)
31 \( 1 + (-7.49 + 2.00i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.18 + 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.16 - 1.16i)T - 41iT^{2} \)
43 \( 1 - 1.15iT - 43T^{2} \)
47 \( 1 + (2.55 - 9.54i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.17 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (12.1 - 3.26i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.96 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 - 2.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.54 + 1.54i)T - 71iT^{2} \)
73 \( 1 + (-0.439 - 1.64i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.34 + 5.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.17 + 4.17i)T - 83iT^{2} \)
89 \( 1 + (-0.853 + 3.18i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.14 - 5.14i)T + 97iT^{2} \)
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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

−10.93487706592235046492522427663, −10.05189801160717879131119685341, −8.806599689503111444370338565350, −7.999495544744754195102807115471, −7.57801667931246227689208296447, −6.34130937965414527106252331132, −4.74021475255710186794263774238, −4.34650256372823699198808672130, −3.19016318828789282585489299625, −1.82975067851193327153217981430, 1.63575566245610979094001297751, 3.00738037322769532218657829720, 3.82001191958813796293155102143, 4.90618896940202982828748645847, 6.05583779020460414181171120982, 7.44665532249335466831044550045, 7.959234415602862874245593820904, 8.682049361965677947381971625067, 10.06024079742514308614403454254, 10.80891323779974876559966395396

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