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Properties

Label 2-546-273.251-c1-0-8
Degree $2$
Conductor $546$
Sign $0.225 - 0.974i$
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  + (−0.5 + 0.866i)2-s + (−1.68 + 0.396i)3-s + (−0.499 − 0.866i)4-s + 2.52i·5-s + (0.5 − 1.65i)6-s + (0.5 − 2.59i)7-s + 0.999·8-s + (2.68 − 1.33i)9-s + (−2.18 − 1.26i)10-s + (0.686 − 1.18i)11-s + (1.18 + 1.26i)12-s + (3.5 − 0.866i)13-s + (2 + 1.73i)14-s + (−1 − 4.25i)15-s + (−0.5 + 0.866i)16-s + (0.686 + 1.18i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.973 + 0.228i)3-s + (−0.249 − 0.433i)4-s + 1.12i·5-s + (0.204 − 0.677i)6-s + (0.188 − 0.981i)7-s + 0.353·8-s + (0.895 − 0.445i)9-s + (−0.691 − 0.399i)10-s + (0.206 − 0.358i)11-s + (0.342 + 0.364i)12-s + (0.970 − 0.240i)13-s + (0.534 + 0.462i)14-s + (−0.258 − 1.09i)15-s + (−0.125 + 0.216i)16-s + (0.166 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.225 - 0.974i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715129 + 0.568438i\)
\(L(\frac12)\) \(\approx\) \(0.715129 + 0.568438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.68 - 0.396i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 - 2.52iT - 5T^{2} \)
11 \( 1 + (-0.686 + 1.18i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.686 - 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.68 - 2.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.813 + 0.469i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.686 - 0.396i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (-7.11 - 4.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.31 - 3.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.25iT - 47T^{2} \)
53 \( 1 - 14.3iT - 53T^{2} \)
59 \( 1 + (-8.18 + 4.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.11 + 4.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.558 + 0.967i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.744T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 5.04iT - 83T^{2} \)
89 \( 1 + (10.8 + 6.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.37 - 9.30i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.89123321090293402200688152329, −10.29563500302432838184336049602, −9.430067939550606446889747315100, −8.031735233434157750492183280665, −7.28014965911326438153686602871, −6.37235885148769608984920131509, −5.83583070621608656820320078046, −4.43472869959853480670515477462, −3.44348174740318559275163512851, −1.13356926218376750695888098837, 0.896157220491492238340477353492, 2.13018686585542998227641143854, 4.00255217071665827496674906336, 5.05634268518439224211286119090, 5.75890650243240381233705541885, 7.00909713009274810126302260910, 8.167764446971837822182160391918, 9.014572319086605493814157745034, 9.633969688758369843260456658267, 10.84963928336128222337997898997

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