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Properties

Label 2-548-548.119-c0-0-0
Degree $2$
Conductor $548$
Sign $0.783 + 0.621i$
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 + (0.172 + 0.0666i)5-s + (0.445 − 0.895i)8-s + (−0.850 − 0.526i)9-s + 0.184·10-s + (−0.890 + 1.17i)13-s + (0.0922 − 0.995i)16-s + (0.397 + 0.798i)17-s + (−0.982 − 0.183i)18-s + (0.172 − 0.0666i)20-s + (−0.713 − 0.650i)25-s + (−0.404 + 1.42i)26-s + (−0.111 + 1.20i)29-s + (−0.273 − 0.961i)32-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (0.739 − 0.673i)4-s + (0.172 + 0.0666i)5-s + (0.445 − 0.895i)8-s + (−0.850 − 0.526i)9-s + 0.184·10-s + (−0.890 + 1.17i)13-s + (0.0922 − 0.995i)16-s + (0.397 + 0.798i)17-s + (−0.982 − 0.183i)18-s + (0.172 − 0.0666i)20-s + (−0.713 − 0.650i)25-s + (−0.404 + 1.42i)26-s + (−0.111 + 1.20i)29-s + (−0.273 − 0.961i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.783 + 0.621i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448418614\)
\(L(\frac12)\) \(\approx\) \(1.448418614\)
\(L(1)\) 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 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
7 \( 1 + (0.273 + 0.961i)T^{2} \)
11 \( 1 + (-0.0922 + 0.995i)T^{2} \)
13 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
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 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
31 \( 1 + (-0.932 + 0.361i)T^{2} \)
37 \( 1 - 1.86T + T^{2} \)
41 \( 1 + 1.96T + T^{2} \)
43 \( 1 + (-0.932 - 0.361i)T^{2} \)
47 \( 1 + (-0.445 - 0.895i)T^{2} \)
53 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
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 \( 1 + (0.510 + 0.197i)T + (0.739 + 0.673i)T^{2} \)
97 \( 1 + (-1.47 + 1.34i)T + (0.0922 - 0.995i)T^{2} \)
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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

−11.19754312574466037508816194112, −10.11576936207512048750861648537, −9.420751572534776995729636694729, −8.242664621308856742224576494630, −6.96319685591388291193142921624, −6.23012276478978690516223590136, −5.27515405510847908360053239417, −4.20696231842705774590952938580, −3.12568774804216882472431789574, −1.90237035352063489059094852594, 2.39021367274351185788840095975, 3.29628334118178233557636111421, 4.76670010815105455193294622455, 5.45844738517858117144961398666, 6.30381348325078337469736651657, 7.68366002729006110566585482724, 7.950681741587654226792093113717, 9.361446691080696973855496610587, 10.36831057310977043683871689762, 11.38185266411809725450203823839

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