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Properties

Label 2-548-548.407-c0-0-0
Degree $2$
Conductor $548$
Sign $0.999 - 0.00647i$
Analytic cond. $0.273487$
Root an. cond. $0.522960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0922 + 0.995i)2-s + (−0.982 + 0.183i)4-s + (0.172 − 1.85i)5-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + 1.86·10-s + (1.67 + 1.03i)13-s + (0.932 − 0.361i)16-s + (0.149 − 0.526i)17-s + (0.739 − 0.673i)18-s + (0.172 + 1.85i)20-s + (−2.43 − 0.455i)25-s + (−0.876 + 1.75i)26-s + (−1.58 + 0.614i)29-s + (0.445 + 0.895i)32-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (−0.982 + 0.183i)4-s + (0.172 − 1.85i)5-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + 1.86·10-s + (1.67 + 1.03i)13-s + (0.932 − 0.361i)16-s + (0.149 − 0.526i)17-s + (0.739 − 0.673i)18-s + (0.172 + 1.85i)20-s + (−2.43 − 0.455i)25-s + (−0.876 + 1.75i)26-s + (−1.58 + 0.614i)29-s + (0.445 + 0.895i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.999 - 0.00647i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.999 - 0.00647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8580328703\)
\(L(\frac12)\) \(\approx\) \(0.8580328703\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0922 - 0.995i)T \)
137 \( 1 + (0.850 - 0.526i)T \)
good3 \( 1 + (0.602 + 0.798i)T^{2} \)
5 \( 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2} \)
7 \( 1 + (-0.445 - 0.895i)T^{2} \)
11 \( 1 + (-0.932 + 0.361i)T^{2} \)
13 \( 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2} \)
17 \( 1 + (-0.149 + 0.526i)T + (-0.850 - 0.526i)T^{2} \)
19 \( 1 + (-0.0922 + 0.995i)T^{2} \)
23 \( 1 + (-0.739 + 0.673i)T^{2} \)
29 \( 1 + (1.58 - 0.614i)T + (0.739 - 0.673i)T^{2} \)
31 \( 1 + (-0.0922 - 0.995i)T^{2} \)
37 \( 1 - 0.184T + T^{2} \)
41 \( 1 - 1.47T + T^{2} \)
43 \( 1 + (-0.0922 + 0.995i)T^{2} \)
47 \( 1 + (0.273 - 0.961i)T^{2} \)
53 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
59 \( 1 + (0.273 - 0.961i)T^{2} \)
61 \( 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2} \)
67 \( 1 + (-0.445 - 0.895i)T^{2} \)
71 \( 1 + (-0.932 - 0.361i)T^{2} \)
73 \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \)
79 \( 1 + (0.602 - 0.798i)T^{2} \)
83 \( 1 + (0.850 - 0.526i)T^{2} \)
89 \( 1 + (-0.0822 + 0.887i)T + (-0.982 - 0.183i)T^{2} \)
97 \( 1 + (1.96 - 0.367i)T + (0.932 - 0.361i)T^{2} \)
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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

−11.16293103798721449716459383663, −9.449851895653138246128429142861, −9.081519386603755910645137381057, −8.553456548991361460714249116904, −7.50937463139777613337186662523, −6.16109302869371092377969477177, −5.65926748268151989379999936564, −4.53257381122351095209532148341, −3.72750736722684338405898308070, −1.17889893243445309532144144878, 2.07523842038624971016352659661, 3.11608428742277406107411929512, 3.83506800268964382062193303635, 5.59229289011405905044441801408, 6.16336298111157457058458905595, 7.63217201261109887376283059756, 8.353430377482078210251844111147, 9.605325274255010663384413296135, 10.49317590926347078372302021956, 11.07473990267820381349237626325

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