Properties

Label 2-548-548.323-c0-0-0
Degree $2$
Conductor $548$
Sign $0.885 - 0.464i$
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  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.840 + 1.35i)5-s + (0.0922 − 0.995i)8-s + (−0.739 − 0.673i)9-s − 1.59i·10-s + (0.328 + 1.75i)13-s + (−0.602 + 0.798i)16-s + (−0.0170 − 0.183i)17-s + (0.273 + 0.961i)18-s + (−0.840 + 1.35i)20-s + (−0.689 + 1.38i)25-s + (0.646 − 1.66i)26-s + (−0.293 − 0.221i)29-s + (0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.840 + 1.35i)5-s + (0.0922 − 0.995i)8-s + (−0.739 − 0.673i)9-s − 1.59i·10-s + (0.328 + 1.75i)13-s + (−0.602 + 0.798i)16-s + (−0.0170 − 0.183i)17-s + (0.273 + 0.961i)18-s + (−0.840 + 1.35i)20-s + (−0.689 + 1.38i)25-s + (0.646 − 1.66i)26-s + (−0.293 − 0.221i)29-s + (0.932 − 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.885 - 0.464i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.885 - 0.464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6700364059\)
\(L(\frac12)\) \(\approx\) \(0.6700364059\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.850 + 0.526i)T \)
137 \( 1 + (0.982 + 0.183i)T \)
good3 \( 1 + (0.739 + 0.673i)T^{2} \)
5 \( 1 + (-0.840 - 1.35i)T + (-0.445 + 0.895i)T^{2} \)
7 \( 1 + (-0.932 + 0.361i)T^{2} \)
11 \( 1 + (0.602 - 0.798i)T^{2} \)
13 \( 1 + (-0.328 - 1.75i)T + (-0.932 + 0.361i)T^{2} \)
17 \( 1 + (0.0170 + 0.183i)T + (-0.982 + 0.183i)T^{2} \)
19 \( 1 + (0.850 - 0.526i)T^{2} \)
23 \( 1 + (-0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.293 + 0.221i)T + (0.273 + 0.961i)T^{2} \)
31 \( 1 + (-0.850 - 0.526i)T^{2} \)
37 \( 1 - 1.70T + T^{2} \)
41 \( 1 + 1.92iT - T^{2} \)
43 \( 1 + (-0.850 + 0.526i)T^{2} \)
47 \( 1 + (0.0922 + 0.995i)T^{2} \)
53 \( 1 + (1.91 - 0.544i)T + (0.850 - 0.526i)T^{2} \)
59 \( 1 + (-0.0922 - 0.995i)T^{2} \)
61 \( 1 + (-0.890 + 0.811i)T + (0.0922 - 0.995i)T^{2} \)
67 \( 1 + (0.932 - 0.361i)T^{2} \)
71 \( 1 + (-0.602 - 0.798i)T^{2} \)
73 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
79 \( 1 + (0.739 - 0.673i)T^{2} \)
83 \( 1 + (-0.982 - 0.183i)T^{2} \)
89 \( 1 + (0.380 + 0.614i)T + (-0.445 + 0.895i)T^{2} \)
97 \( 1 + (0.602 - 0.798i)T^{2} \)
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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

−11.18058820909523949068785189955, −10.12877621666011497530002594106, −9.392774526904410036746425679966, −8.784679185094675738591364654934, −7.43112055099690969734306541320, −6.63786083098973026793296470090, −5.99620527657974655649059667984, −4.01925440321343188875852504006, −2.91837561656905506193056451694, −1.95393220838821908376445896495, 1.18479807552869919182849643118, 2.66048684482672471279540711450, 4.81408633647897477800046911829, 5.57477544018274058569533984763, 6.16098985205560527034481358957, 7.81531766858364879549506522071, 8.241294635414856603133360911199, 9.093627819433070837656625601907, 9.881292350606963497660091697272, 10.68054576447842248306727932082

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