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Properties

Label 2-546-91.9-c1-0-6
Degree $2$
Conductor $546$
Sign $-0.345 - 0.938i$
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 + 3-s + (−0.499 + 0.866i)4-s + (−1.15 + 1.99i)5-s + (0.5 + 0.866i)6-s + (2.61 + 0.396i)7-s − 0.999·8-s + 9-s − 2.30·10-s − 0.357·11-s + (−0.499 + 0.866i)12-s + (−2.73 + 2.34i)13-s + (0.964 + 2.46i)14-s + (−1.15 + 1.99i)15-s + (−0.5 − 0.866i)16-s + (−2.12 + 3.67i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 + 0.433i)4-s + (−0.514 + 0.891i)5-s + (0.204 + 0.353i)6-s + (0.988 + 0.149i)7-s − 0.353·8-s + 0.333·9-s − 0.728·10-s − 0.107·11-s + (−0.144 + 0.249i)12-s + (−0.759 + 0.650i)13-s + (0.257 + 0.658i)14-s + (−0.297 + 0.514i)15-s + (−0.125 − 0.216i)16-s + (−0.515 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10955 + 1.59005i\)
\(L(\frac12)\) \(\approx\) \(1.10955 + 1.59005i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-2.61 - 0.396i)T \)
13 \( 1 + (2.73 - 2.34i)T \)
good5 \( 1 + (1.15 - 1.99i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.357T + 11T^{2} \)
17 \( 1 + (2.12 - 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 6.79T + 19T^{2} \)
23 \( 1 + (1.77 + 3.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.624 - 1.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.124 + 0.214i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.72 + 8.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.95 - 5.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.63 - 4.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.19 + 7.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.605 + 1.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.58 + 6.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + 2.81T + 67T^{2} \)
71 \( 1 + (-0.794 - 1.37i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.407 - 0.706i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.77 + 9.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 + (0.367 + 0.636i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.63 + 6.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.19588580742518484367193600418, −10.18074706976737056257734657537, −9.077745154601427486329118183462, −8.193163213697632253199891948856, −7.43021097905931163376156633999, −6.80943637710442170106798404268, −5.46395807881328567028370508827, −4.42362704592469627942022458912, −3.42507834996319464686848945119, −2.14766218060726463402465887765, 1.02118599518658934194148192329, 2.48742666671619081066581039713, 3.74781453901634895254998468853, 4.84272414434830469464318557303, 5.33786747634494046088436430960, 7.22487078999334766595388339893, 7.939757398445845615839659427995, 8.812053156952799511838268254062, 9.653580927429298786330923320335, 10.55711539503727233203587347125

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