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Properties

Label 2-546-273.251-c1-0-21
Degree $2$
Conductor $546$
Sign $-0.593 + 0.805i$
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 + (−1.68 − 0.405i)3-s + (−0.499 − 0.866i)4-s − 0.465i·5-s + (−1.19 + 1.25i)6-s + (2.62 + 0.293i)7-s − 0.999·8-s + (2.67 + 1.36i)9-s + (−0.403 − 0.232i)10-s + (1.39 − 2.41i)11-s + (0.490 + 1.66i)12-s + (−0.799 − 3.51i)13-s + (1.56 − 2.13i)14-s + (−0.189 + 0.784i)15-s + (−0.5 + 0.866i)16-s + (0.508 + 0.880i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.972 − 0.234i)3-s + (−0.249 − 0.433i)4-s − 0.208i·5-s + (−0.487 + 0.512i)6-s + (0.993 + 0.111i)7-s − 0.353·8-s + (0.890 + 0.455i)9-s + (−0.127 − 0.0736i)10-s + (0.421 − 0.729i)11-s + (0.141 + 0.479i)12-s + (−0.221 − 0.975i)13-s + (0.419 − 0.569i)14-s + (−0.0488 + 0.202i)15-s + (−0.125 + 0.216i)16-s + (0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.593 + 0.805i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.593 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562234 - 1.11260i\)
\(L(\frac12)\) \(\approx\) \(0.562234 - 1.11260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.405i)T \)
7 \( 1 + (-2.62 - 0.293i)T \)
13 \( 1 + (0.799 + 3.51i)T \)
good5 \( 1 + 0.465iT - 5T^{2} \)
11 \( 1 + (-1.39 + 2.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.508 - 0.880i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.817 + 1.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.14 + 4.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.01 + 2.31i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.94T + 31T^{2} \)
37 \( 1 + (-4.51 - 2.60i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.35 + 5.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.24 + 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.277iT - 47T^{2} \)
53 \( 1 - 8.99iT - 53T^{2} \)
59 \( 1 + (-8.80 + 5.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.2 + 6.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.308 + 0.178i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.01 - 8.68i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.46T + 73T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 + (-10.1 - 5.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.82 - 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.65501271499781739582299752954, −10.05451431290319283944604508819, −8.666937923636250313477103759608, −7.898144145297481975297035177312, −6.58980040608116277361928962568, −5.62927963004632112313530622312, −4.92304176033985529060011854013, −3.86402322094891165865855350773, −2.17371542111308941395023940177, −0.77215408386970590445452947163, 1.74419976126018144853664846976, 3.88385949952360319973593943039, 4.65708103343727448107254589395, 5.49396673218834615770747088567, 6.58619860636514912538441525989, 7.23632133186003337451845028629, 8.251055989409595767646537276292, 9.472934271571384159166161297506, 10.21103899309857674070632798811, 11.46709431957963754851600541431

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