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Properties

Label 2-546-273.242-c1-0-25
Degree $2$
Conductor $546$
Sign $0.959 - 0.280i$
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 + (1.59 − 0.673i)3-s + (0.866 + 0.499i)4-s + (0.579 + 0.155i)5-s + (1.71 − 0.237i)6-s + (0.0183 + 2.64i)7-s + (0.707 + 0.707i)8-s + (2.09 − 2.15i)9-s + (0.519 + 0.299i)10-s + (−3.10 + 0.831i)11-s + (1.71 + 0.214i)12-s + (3.60 + 0.0973i)13-s + (−0.666 + 2.56i)14-s + (1.02 − 0.142i)15-s + (0.500 + 0.866i)16-s + (−2.59 + 4.49i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.921 − 0.389i)3-s + (0.433 + 0.249i)4-s + (0.259 + 0.0694i)5-s + (0.700 − 0.0971i)6-s + (0.00694 + 0.999i)7-s + (0.249 + 0.249i)8-s + (0.697 − 0.716i)9-s + (0.164 + 0.0948i)10-s + (−0.935 + 0.250i)11-s + (0.496 + 0.0618i)12-s + (0.999 + 0.0270i)13-s + (−0.178 + 0.684i)14-s + (0.265 − 0.0368i)15-s + (0.125 + 0.216i)16-s + (−0.629 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.959 - 0.280i$
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: \((2,\ 546,\ (\ :1/2),\ 0.959 - 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.92198 + 0.417673i\)
\(L(\frac12)\) \(\approx\) \(2.92198 + 0.417673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 + 0.673i)T \)
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

−10.99661654763354350815680447907, −9.845648119676207455035544904568, −8.822939886469849589127553854831, −8.254912099746135201717091408006, −7.16248093348703694031167403491, −6.25919604473886040102708243234, −5.31306867156441396269386817576, −4.04221445855943778965798325871, −2.84092810897801904147056226966, −2.00019542408958651434982427935, 1.65665552706137933120451254581, 3.11058390622078264514371247853, 3.87766664819117652016250125622, 4.93011759632220141763179209714, 5.99655947101236316383952693171, 7.33822137766223710703284072461, 7.939937623383469133830069885211, 9.105504869758761933603917074259, 10.04357775892648007726417297407, 10.67325921077235196583731764678

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