Properties

Label 2-546-13.12-c1-0-6
Degree $2$
Conductor $546$
Sign $0.155 + 0.987i$
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 − 3-s − 4-s + 0.561i·5-s + i·6-s + i·7-s + i·8-s + 9-s + 0.561·10-s − 1.43i·11-s + 12-s + (−0.561 − 3.56i)13-s + 14-s − 0.561i·15-s + 16-s + 5.68·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.251i·5-s + 0.408i·6-s + 0.377i·7-s + 0.353i·8-s + 0.333·9-s + 0.177·10-s − 0.433i·11-s + 0.288·12-s + (−0.155 − 0.987i)13-s + 0.267·14-s − 0.144i·15-s + 0.250·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846329 - 0.723343i\)
\(L(\frac12)\) \(\approx\) \(0.846329 - 0.723343i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 - iT \)
13 \( 1 + (0.561 + 3.56i)T \)
good5 \( 1 - 0.561iT - 5T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
17 \( 1 - 5.68T + 17T^{2} \)
19 \( 1 + 2.56iT - 19T^{2} \)
23 \( 1 - 5.68T + 23T^{2} \)
29 \( 1 + 2.56T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 - 1.68iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + 6.24iT - 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 2.56T + 61T^{2} \)
67 \( 1 + 7.12iT - 67T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + 7.43iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.65293478224083194080861743413, −9.989144793989666392904901866589, −9.010592278323758335246929041063, −8.010271066408051942881125907744, −6.97364234987324441591874111064, −5.69016212473853088460661458379, −5.10285295172301647332546109227, −3.63539967962175318920647713713, −2.62615273599389016373788112315, −0.834707819000932310465833839667, 1.31355843075987360880619276384, 3.46453253255970694580335035908, 4.71893763562569546200998891954, 5.36164933279997614873877654856, 6.61875524705064131753866320282, 7.17424229232181877757388618748, 8.223207276587349189906413582017, 9.233520644683733315640114958847, 10.04087424549474826959752052262, 10.91252293023656831853274305266

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