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Properties

Label 2-546-21.17-c1-0-18
Degree $2$
Conductor $546$
Sign $0.555 + 0.831i$
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 + (0.587 − 1.62i)3-s + (0.499 − 0.866i)4-s + (0.386 + 0.669i)5-s + (0.306 + 1.70i)6-s + (1.82 − 1.91i)7-s + 0.999i·8-s + (−2.31 − 1.91i)9-s + (−0.669 − 0.386i)10-s + (3.31 + 1.91i)11-s + (−1.11 − 1.32i)12-s − i·13-s + (−0.627 + 2.57i)14-s + (1.31 − 0.236i)15-s + (−0.5 − 0.866i)16-s + (0.460 − 0.797i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.339 − 0.940i)3-s + (0.249 − 0.433i)4-s + (0.172 + 0.299i)5-s + (0.124 + 0.695i)6-s + (0.691 − 0.722i)7-s + 0.353i·8-s + (−0.770 − 0.637i)9-s + (−0.211 − 0.122i)10-s + (0.999 + 0.576i)11-s + (−0.322 − 0.382i)12-s − 0.277i·13-s + (−0.167 + 0.686i)14-s + (0.340 − 0.0610i)15-s + (−0.125 − 0.216i)16-s + (0.111 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.555 + 0.831i$
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: \((2,\ 546,\ (\ :1/2),\ 0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18761 - 0.635047i\)
\(L(\frac12)\) \(\approx\) \(1.18761 - 0.635047i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 1.62i)T \)
7 \( 1 + (-1.82 + 1.91i)T \)
13 \( 1 + iT \)
good5 \( 1 + (-0.386 - 0.669i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.31 - 1.91i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.460 + 0.797i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.92 + 1.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.80 - 2.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.669iT - 29T^{2} \)
31 \( 1 + (-0.824 - 0.475i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 + 5.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.36T + 41T^{2} \)
43 \( 1 - 1.18T + 43T^{2} \)
47 \( 1 + (5.17 + 8.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.30 - 4.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.358 - 0.207i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.70 - 8.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.24iT - 71T^{2} \)
73 \( 1 + (-7.63 - 4.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.47 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.22T + 83T^{2} \)
89 \( 1 + (-2.00 - 3.48i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.41iT - 97T^{2} \)
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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

−10.57067868539274057908249731171, −9.651170043577024848248482424629, −8.752214731552335615082370481406, −7.85437694692080428423818013418, −7.15141661083880224900046716793, −6.50824376983729519090327807164, −5.29343320860356202123739393286, −3.80514878785079773043027855798, −2.22999335299966813410017565924, −1.03955522277227443757726973694, 1.63174007043946116620705964588, 3.02941841931400028383690204229, 4.14028132600387706584398319058, 5.24423011774837570325900815043, 6.27829151769930758728939648092, 7.80765456519035146622148896251, 8.583431658899147152176436076553, 9.167463245464428125928007110175, 9.896894457728376330072746764906, 10.89411070477430215124694891341

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