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Properties

Label 2-546-21.20-c1-0-16
Degree $2$
Conductor $546$
Sign $0.487 - 0.872i$
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  + i·2-s + (1.61 + 0.618i)3-s − 4-s + 1.23·5-s + (−0.618 + 1.61i)6-s + (0.381 − 2.61i)7-s − i·8-s + (2.23 + 2.00i)9-s + 1.23i·10-s + (−1.61 − 0.618i)12-s + i·13-s + (2.61 + 0.381i)14-s + (2.00 + 0.763i)15-s + 16-s + 6.47·17-s + (−2.00 + 2.23i)18-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.934 + 0.356i)3-s − 0.5·4-s + 0.552·5-s + (−0.252 + 0.660i)6-s + (0.144 − 0.989i)7-s − 0.353i·8-s + (0.745 + 0.666i)9-s + 0.390i·10-s + (−0.467 − 0.178i)12-s + 0.277i·13-s + (0.699 + 0.102i)14-s + (0.516 + 0.197i)15-s + 0.250·16-s + 1.56·17-s + (−0.471 + 0.527i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.487 - 0.872i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83444 + 1.07613i\)
\(L(\frac12)\) \(\approx\) \(1.83444 + 1.07613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.61 - 0.618i)T \)
7 \( 1 + (-0.381 + 2.61i)T \)
13 \( 1 - iT \)
good5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 - 0.763T + 41T^{2} \)
43 \( 1 + 1.52T + 43T^{2} \)
47 \( 1 + 2.47T + 47T^{2} \)
53 \( 1 + 1.23iT - 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 4.47iT - 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 5.52iT - 71T^{2} \)
73 \( 1 - 1.23iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 5.23iT - 97T^{2} \)
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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

−10.37278421775053452799056430433, −10.05015353347892694308573278316, −9.202751338564079190143547414234, −7.977941142261668178925631006858, −7.70331423151944173814889426918, −6.46998500883022625827900114357, −5.39117101119505003622801371273, −4.22955413460248015073444137335, −3.38034725442819305460963527797, −1.62945435596401620239739031992, 1.48097663912574349043956344955, 2.60846475487949686322216879101, 3.43546159288879276435277033172, 4.96402759332817801974993888012, 5.92313893828954916274215533591, 7.24248859219790444685153162132, 8.218945276446160913730699923988, 9.059601812687755749320191724844, 9.613302918717973054983561467944, 10.52030581981540503751361107006

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