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Properties

Label 2-546-273.272-c1-0-8
Degree $2$
Conductor $546$
Sign $0.982 - 0.185i$
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 + (0.420 − 1.68i)3-s + 4-s + 3.36i·5-s + (−0.420 + 1.68i)6-s + (−2.37 − 1.16i)7-s − 8-s + (−2.64 − 1.41i)9-s − 3.36i·10-s + (0.420 − 1.68i)12-s + (2.79 + 2.27i)13-s + (2.37 + 1.16i)14-s + (5.64 + 1.41i)15-s + 16-s + 7.82·17-s + (2.64 + 1.41i)18-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.242 − 0.970i)3-s + 0.5·4-s + 1.50i·5-s + (−0.171 + 0.685i)6-s + (−0.898 − 0.439i)7-s − 0.353·8-s + (−0.881 − 0.471i)9-s − 1.06i·10-s + (0.121 − 0.485i)12-s + (0.775 + 0.631i)13-s + (0.635 + 0.311i)14-s + (1.45 + 0.365i)15-s + 0.250·16-s + 1.89·17-s + (0.623 + 0.333i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.982 - 0.185i$
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: \((2,\ 546,\ (\ :1/2),\ 0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05594 + 0.0990224i\)
\(L(\frac12)\) \(\approx\) \(1.05594 + 0.0990224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.420 + 1.68i)T \)
7 \( 1 + (2.37 + 1.16i)T \)
13 \( 1 + (-2.79 - 2.27i)T \)
good5 \( 1 - 3.36iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 - 5.59T + 19T^{2} \)
23 \( 1 - 0.500iT - 23T^{2} \)
29 \( 1 - 5.15iT - 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 + 2.32iT - 37T^{2} \)
41 \( 1 - 9.87iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 4.33iT - 47T^{2} \)
53 \( 1 - 0.500iT - 53T^{2} \)
59 \( 1 + 2.16iT - 59T^{2} \)
61 \( 1 + 4.55iT - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 + 6.58T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 0.708T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 3.14iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.76664186126293810101830179906, −9.916475443116376499785219882987, −9.167031318167278057008508814149, −7.80074191651352830637297301512, −7.35646044980610061339068807573, −6.51449242067776425185923247844, −5.87962341431085685903643453095, −3.40214944337826470161130702305, −2.95352209525819191065984164038, −1.26197729513931330076078937803, 0.915252312202299965815811968374, 2.92023749726845299461940628308, 3.94724809318322377748231286009, 5.43716905281000407551577407652, 5.77256102194937356589801830755, 7.59075754957123330499903252141, 8.438119577035053085038084883833, 9.054282386877216681764366050396, 9.818102760941267189415381276219, 10.30362356603257389721269459020

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