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Properties

Label 2-546-273.272-c1-0-12
Degree $2$
Conductor $546$
Sign $0.977 - 0.209i$
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  + 2-s + (−1.68 − 0.420i)3-s + 4-s − 0.841i·5-s + (−1.68 − 0.420i)6-s + (−0.595 + 2.57i)7-s + 8-s + (2.64 + 1.41i)9-s − 0.841i·10-s + (−1.68 − 0.420i)12-s + (2.27 + 2.79i)13-s + (−0.595 + 2.57i)14-s + (−0.354 + 1.41i)15-s + 16-s + 4.33·17-s + (2.64 + 1.41i)18-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.970 − 0.242i)3-s + 0.5·4-s − 0.376i·5-s + (−0.685 − 0.171i)6-s + (−0.224 + 0.974i)7-s + 0.353·8-s + (0.881 + 0.471i)9-s − 0.266i·10-s + (−0.485 − 0.121i)12-s + (0.631 + 0.775i)13-s + (−0.159 + 0.688i)14-s + (−0.0914 + 0.365i)15-s + 0.250·16-s + 1.05·17-s + (0.623 + 0.333i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70790 + 0.180492i\)
\(L(\frac12)\) \(\approx\) \(1.70790 + 0.180492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.68 + 0.420i)T \)
7 \( 1 + (0.595 - 2.57i)T \)
13 \( 1 + (-2.27 - 2.79i)T \)
good5 \( 1 + 0.841iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 4.33T + 17T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + 7.98iT - 23T^{2} \)
29 \( 1 - 2.32iT - 29T^{2} \)
31 \( 1 + 5.53T + 31T^{2} \)
37 \( 1 - 5.15iT - 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 7.82iT - 47T^{2} \)
53 \( 1 + 7.98iT - 53T^{2} \)
59 \( 1 + 3.91iT - 59T^{2} \)
61 \( 1 + 5.59iT - 61T^{2} \)
67 \( 1 - 1.82iT - 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 - 3.14T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 7.27iT - 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + 9.87T + 97T^{2} \)
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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

−11.18779688222115683795239360309, −10.12238474809705879191999563567, −9.130588867952283654573955095847, −8.030120801685852215529100833837, −6.85288413432545398108540662690, −6.09421568401466517916372460180, −5.29253576279677211842792451388, −4.45865238423089957367609717624, −2.98843241552799605161279797847, −1.39773180230670617994151553922, 1.11320497805566594274401066309, 3.31159540622794470134586252896, 3.99643544585557338642938649104, 5.40340275125346878666517430381, 5.83599527612380480984088413203, 7.21034172671887663500814593469, 7.49336755883354975339983008925, 9.300333592214685633026706964239, 10.32320644570532475168544384539, 10.75485635114454952083858074915

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