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Properties

Label 2-546-13.10-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.967 - 0.252i$
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 + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 3.46i·5-s + (0.866 + 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.73 − 2.99i)10-s + (−3.46 + 2i)11-s − 0.999·12-s + (−2.59 − 2.5i)13-s − 0.999·14-s + (2.99 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−0.232 + 0.401i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 1.54i·5-s + (0.353 + 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.547 − 0.948i)10-s + (−1.04 + 0.603i)11-s − 0.288·12-s + (−0.720 − 0.693i)13-s − 0.267·14-s + (0.774 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.0562 + 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.967 - 0.252i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0548414 + 0.427139i\)
\(L(\frac12)\) \(\approx\) \(0.0548414 + 0.427139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + (3.46 - 2i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.232 - 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.464 + 0.267i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 - 1.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.464iT - 31T^{2} \)
37 \( 1 + (7.73 - 4.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.23 - 2.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.46iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-8.42 - 4.86i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.59 - 4.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.76 + 1.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.3 - 5.96i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.92iT - 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 + (-7.16 + 4.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.26 - 4.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.02920433051573234781770880336, −10.31746455422318511450067967974, −9.693204269498469869828276721991, −8.175850886116749958113091295374, −7.56608541158320025189730485218, −6.89939411990615936289009071744, −5.95253303556012599295410803858, −4.95476999836256925769902203774, −3.04019460697396748757509041738, −2.07883979412458789837696819307, 0.29548496073519443075110509941, 1.90383509974229487257351589042, 3.61383730384312203796128503229, 4.88049162606403145839739213054, 5.34705187045503085558831304047, 6.93224731100799593607005033012, 8.076422082364077352627368096287, 8.761789357216632650964894984835, 9.424914473979546242573184794845, 10.39599908396827466536238915123

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