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Properties

Label 2-546-21.17-c1-0-6
Degree $2$
Conductor $546$
Sign $0.711 - 0.702i$
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  + (−0.866 + 0.5i)2-s + (−1.70 − 0.321i)3-s + (0.499 − 0.866i)4-s + (1.63 + 2.83i)5-s + (1.63 − 0.572i)6-s + (1.76 − 1.96i)7-s + 0.999i·8-s + (2.79 + 1.09i)9-s + (−2.83 − 1.63i)10-s + (−0.671 − 0.387i)11-s + (−1.12 + 1.31i)12-s − i·13-s + (−0.545 + 2.58i)14-s + (−1.87 − 5.36i)15-s + (−0.5 − 0.866i)16-s + (0.0317 − 0.0550i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.982 − 0.185i)3-s + (0.249 − 0.433i)4-s + (0.733 + 1.27i)5-s + (0.667 − 0.233i)6-s + (0.667 − 0.744i)7-s + 0.353i·8-s + (0.931 + 0.364i)9-s + (−0.898 − 0.518i)10-s + (−0.202 − 0.116i)11-s + (−0.326 + 0.379i)12-s − 0.277i·13-s + (−0.145 + 0.691i)14-s + (−0.484 − 1.38i)15-s + (−0.125 − 0.216i)16-s + (0.00770 − 0.0133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900966 + 0.370009i\)
\(L(\frac12)\) \(\approx\) \(0.900966 + 0.370009i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (1.70 + 0.321i)T \)
7 \( 1 + (-1.76 + 1.96i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-1.63 - 2.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.671 + 0.387i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0317 + 0.0550i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.489 - 0.282i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.86 + 3.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.95iT - 29T^{2} \)
31 \( 1 + (-0.453 - 0.262i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.48 - 7.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.420T + 41T^{2} \)
43 \( 1 - 8.44T + 43T^{2} \)
47 \( 1 + (-6.63 - 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.134 - 0.0777i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.93 - 6.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.98 - 1.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.40 + 9.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.904iT - 71T^{2} \)
73 \( 1 + (-12.4 - 7.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.03 + 8.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (2.77 + 4.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.01iT - 97T^{2} \)
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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

−10.76212488954634995881231491596, −10.37026494481594743500761288735, −9.336057503677878022609714098569, −7.965316368942049985559084367148, −7.13021055986912285462077172656, −6.54360202056124250558004838922, −5.63024119696202739299954982789, −4.55949224723841622597198775223, −2.73906681886854362059988683594, −1.19810979948751349224971534174, 0.996421634939569712840327031461, 2.17852410075887812397155935410, 4.23004305558215826359568069018, 5.23486970948018854081153730545, 5.79985595701374731732729111768, 7.13049876335792410870210646163, 8.269126542716426160972882265125, 9.231745409012706484905616285074, 9.590826319339994638200321030832, 10.81124658091979353010852941371

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