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Properties

Label 2-548-548.99-c0-0-0
Degree $2$
Conductor $548$
Sign $0.993 + 0.111i$
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  + (0.932 + 0.361i)2-s + (0.739 + 0.673i)4-s + (−0.719 − 1.85i)5-s + (0.445 + 0.895i)8-s + (0.850 − 0.526i)9-s − 1.99i·10-s + (−1.07 + 0.811i)13-s + (0.0922 + 0.995i)16-s + (−0.397 + 0.798i)17-s + (0.982 − 0.183i)18-s + (0.719 − 1.85i)20-s + (−2.19 + 1.99i)25-s + (−1.29 + 0.368i)26-s + (1.58 − 0.147i)29-s + (−0.273 + 0.961i)32-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)2-s + (0.739 + 0.673i)4-s + (−0.719 − 1.85i)5-s + (0.445 + 0.895i)8-s + (0.850 − 0.526i)9-s − 1.99i·10-s + (−1.07 + 0.811i)13-s + (0.0922 + 0.995i)16-s + (−0.397 + 0.798i)17-s + (0.982 − 0.183i)18-s + (0.719 − 1.85i)20-s + (−2.19 + 1.99i)25-s + (−1.29 + 0.368i)26-s + (1.58 − 0.147i)29-s + (−0.273 + 0.961i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.993 + 0.111i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.993 + 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380808850\)
\(L(\frac12)\) \(\approx\) \(1.380808850\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.932 - 0.361i)T \)
137 \( 1 + (0.602 - 0.798i)T \)
good3 \( 1 + (-0.850 + 0.526i)T^{2} \)
5 \( 1 + (0.719 + 1.85i)T + (-0.739 + 0.673i)T^{2} \)
7 \( 1 + (0.273 - 0.961i)T^{2} \)
11 \( 1 + (-0.0922 - 0.995i)T^{2} \)
13 \( 1 + (1.07 - 0.811i)T + (0.273 - 0.961i)T^{2} \)
17 \( 1 + (0.397 - 0.798i)T + (-0.602 - 0.798i)T^{2} \)
19 \( 1 + (-0.932 + 0.361i)T^{2} \)
23 \( 1 + (-0.982 + 0.183i)T^{2} \)
29 \( 1 + (-1.58 + 0.147i)T + (0.982 - 0.183i)T^{2} \)
31 \( 1 + (0.932 + 0.361i)T^{2} \)
37 \( 1 + 1.86T + T^{2} \)
41 \( 1 - 0.367iT - T^{2} \)
43 \( 1 + (0.932 - 0.361i)T^{2} \)
47 \( 1 + (0.445 - 0.895i)T^{2} \)
53 \( 1 + (0.328 + 1.75i)T + (-0.932 + 0.361i)T^{2} \)
59 \( 1 + (-0.445 + 0.895i)T^{2} \)
61 \( 1 + (-0.156 - 0.0971i)T + (0.445 + 0.895i)T^{2} \)
67 \( 1 + (-0.273 + 0.961i)T^{2} \)
71 \( 1 + (0.0922 - 0.995i)T^{2} \)
73 \( 1 + (-1.02 + 1.35i)T + (-0.273 - 0.961i)T^{2} \)
79 \( 1 + (-0.850 - 0.526i)T^{2} \)
83 \( 1 + (-0.602 + 0.798i)T^{2} \)
89 \( 1 + (0.694 + 1.79i)T + (-0.739 + 0.673i)T^{2} \)
97 \( 1 + (-0.0922 - 0.995i)T^{2} \)
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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

−11.46811254572728259459991198981, −10.06022549942178135404728228285, −8.992161762529517105632620561067, −8.263674810068826752532253011957, −7.33834717408902682356735123618, −6.39100792204776205081053576638, −5.01045607640368444107418848018, −4.55615069698611602458739674142, −3.68956852443612287990417076115, −1.72080799415024235728214864126, 2.37950187238447177506735821537, 3.13644695962862947105816068218, 4.24988780981602750065710844071, 5.28419452521498537144992038298, 6.73067107628311747639504860923, 7.07327593962252851289600625305, 7.943509992691465080964295990001, 9.855109862353566031578210997811, 10.42113052255314342038369330449, 10.96787673274190413776882169885

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