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Properties

Label 2-546-273.242-c1-0-16
Degree $2$
Conductor $546$
Sign $0.00835 + 0.999i$
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  + (−0.965 − 0.258i)2-s + (−0.214 − 1.71i)3-s + (0.866 + 0.499i)4-s + (−0.579 − 0.155i)5-s + (−0.237 + 1.71i)6-s + (0.0183 + 2.64i)7-s + (−0.707 − 0.707i)8-s + (−2.90 + 0.736i)9-s + (0.519 + 0.299i)10-s + (3.10 − 0.831i)11-s + (0.673 − 1.59i)12-s + (3.60 + 0.0973i)13-s + (0.666 − 2.56i)14-s + (−0.142 + 1.02i)15-s + (0.500 + 0.866i)16-s + (2.59 − 4.49i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.123 − 0.992i)3-s + (0.433 + 0.249i)4-s + (−0.259 − 0.0694i)5-s + (−0.0971 + 0.700i)6-s + (0.00694 + 0.999i)7-s + (−0.249 − 0.249i)8-s + (−0.969 + 0.245i)9-s + (0.164 + 0.0948i)10-s + (0.935 − 0.250i)11-s + (0.194 − 0.460i)12-s + (0.999 + 0.0270i)13-s + (0.178 − 0.684i)14-s + (−0.0368 + 0.265i)15-s + (0.125 + 0.216i)16-s + (0.629 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00835 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00835 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.00835 + 0.999i$
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: \((2,\ 546,\ (\ :1/2),\ 0.00835 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684864 - 0.679167i\)
\(L(\frac12)\) \(\approx\) \(0.684864 - 0.679167i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.214 + 1.71i)T \)
7 \( 1 + (-0.0183 - 2.64i)T \)
13 \( 1 + (-3.60 - 0.0973i)T \)
good5 \( 1 + (0.579 + 0.155i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.10 + 0.831i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.59 + 4.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.15 + 8.02i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.637 + 1.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.10iT - 29T^{2} \)
31 \( 1 + (5.29 - 1.41i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.77 - 1.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.19 + 5.19i)T - 41iT^{2} \)
43 \( 1 + 9.47iT - 43T^{2} \)
47 \( 1 + (1.48 - 5.55i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.48 + 0.855i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.14 + 0.842i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.398 - 0.689i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.42 + 1.72i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.81 + 8.81i)T - 71iT^{2} \)
73 \( 1 + (-1.65 - 6.16i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.17 + 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.60 + 9.60i)T - 83iT^{2} \)
89 \( 1 + (-3.10 + 11.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.09 - 9.09i)T + 97iT^{2} \)
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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

−10.85921156903177199149659700408, −9.212741264429799134047010824577, −8.996039649301366075623441302731, −7.957556321169490335237285447189, −7.04947867568886340108378864665, −6.23431530158491856481708011614, −5.22034757312254358877693221371, −3.39358656180782397973401846535, −2.25232355093804968074633736356, −0.816756049908267957888519709421, 1.37690620197009182253117922490, 3.66119912133595227108072091446, 3.99546005521105125996733452367, 5.68089954677914226076384964627, 6.37626032432752542644489514486, 7.76175172588748062760064855492, 8.274393598201452387671133643623, 9.621950216493530158707845677863, 9.883141066821464302006954739722, 10.94886180741886955308821316758

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