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Properties

Label 2-546-91.47-c1-0-13
Degree $2$
Conductor $546$
Sign $0.721 + 0.692i$
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 + (0.866 − 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.419 − 1.56i)5-s + (0.707 − 0.707i)6-s + (1.56 + 2.13i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.810 − 1.40i)10-s + (1.32 + 0.354i)11-s + (0.5 − 0.866i)12-s + (2.72 + 2.36i)13-s + (2.06 + 1.65i)14-s + (−1.14 − 1.14i)15-s + (0.500 − 0.866i)16-s + (−1.90 − 3.29i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.187 − 0.700i)5-s + (0.288 − 0.288i)6-s + (0.591 + 0.805i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.256 − 0.444i)10-s + (0.398 + 0.106i)11-s + (0.144 − 0.249i)12-s + (0.754 + 0.656i)13-s + (0.551 + 0.442i)14-s + (−0.296 − 0.296i)15-s + (0.125 − 0.216i)16-s + (−0.460 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.44774 - 0.984827i\)
\(L(\frac12)\) \(\approx\) \(2.44774 - 0.984827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
13 \( 1 + (-2.72 - 2.36i)T \)
good5 \( 1 + (0.419 + 1.56i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.32 - 0.354i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.90 + 3.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.448 + 1.67i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.60 + 2.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.0101T + 29T^{2} \)
31 \( 1 + (8.33 + 2.23i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.42 - 5.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.53 - 1.53i)T - 41iT^{2} \)
43 \( 1 - 7.52iT - 43T^{2} \)
47 \( 1 + (2.78 - 0.746i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.77 - 3.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.48 - 5.53i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.00 + 0.579i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.555 + 2.07i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.80 - 7.80i)T + 71iT^{2} \)
73 \( 1 + (0.311 - 1.16i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.13 - 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.61 - 3.61i)T - 83iT^{2} \)
89 \( 1 + (-6.89 + 1.84i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.73 + 5.73i)T - 97iT^{2} \)
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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

−11.09475688655217261829719164972, −9.617417774313494476723528457101, −8.858365090204632619705520427360, −8.170530451798583034790134598462, −6.97459584460280153882416815059, −6.01481616328145171670800357245, −4.87503363495235782942656908344, −4.09131293190542946459437285235, −2.66971170822762243977413586681, −1.50584642732729456104760408652, 1.86462897370982772643074354468, 3.51152548496709770287074168633, 3.90772672417092868370918134653, 5.24581243818092681992936564569, 6.35346642996911998171323813746, 7.32377640706526536638627945802, 8.054370059211422886444966273138, 9.010931171284246727513573744619, 10.45252323597561374497034072963, 10.76028386080165339342643769225

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