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Properties

Label 2-548-548.87-c0-0-0
Degree $2$
Conductor $548$
Sign $0.271 + 0.962i$
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  + (0.739 − 0.673i)2-s + (0.0922 − 0.995i)4-s + (−0.247 + 0.271i)5-s + (−0.602 − 0.798i)8-s + (−0.445 − 0.895i)9-s + 0.367i·10-s + (1.91 − 0.544i)13-s + (−0.982 − 0.183i)16-s + (−0.726 + 0.961i)17-s + (−0.932 − 0.361i)18-s + (0.247 + 0.271i)20-s + (0.0798 + 0.861i)25-s + (1.04 − 1.69i)26-s + (−0.353 + 1.89i)29-s + (−0.850 + 0.526i)32-s + ⋯
L(s)  = 1  + (0.739 − 0.673i)2-s + (0.0922 − 0.995i)4-s + (−0.247 + 0.271i)5-s + (−0.602 − 0.798i)8-s + (−0.445 − 0.895i)9-s + 0.367i·10-s + (1.91 − 0.544i)13-s + (−0.982 − 0.183i)16-s + (−0.726 + 0.961i)17-s + (−0.932 − 0.361i)18-s + (0.247 + 0.271i)20-s + (0.0798 + 0.861i)25-s + (1.04 − 1.69i)26-s + (−0.353 + 1.89i)29-s + (−0.850 + 0.526i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.231900884\)
\(L(\frac12)\) \(\approx\) \(1.231900884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.739 + 0.673i)T \)
137 \( 1 + (0.273 - 0.961i)T \)
good3 \( 1 + (0.445 + 0.895i)T^{2} \)
5 \( 1 + (0.247 - 0.271i)T + (-0.0922 - 0.995i)T^{2} \)
7 \( 1 + (0.850 - 0.526i)T^{2} \)
11 \( 1 + (0.982 + 0.183i)T^{2} \)
13 \( 1 + (-1.91 + 0.544i)T + (0.850 - 0.526i)T^{2} \)
17 \( 1 + (0.726 - 0.961i)T + (-0.273 - 0.961i)T^{2} \)
19 \( 1 + (-0.739 - 0.673i)T^{2} \)
23 \( 1 + (0.932 + 0.361i)T^{2} \)
29 \( 1 + (0.353 - 1.89i)T + (-0.932 - 0.361i)T^{2} \)
31 \( 1 + (0.739 - 0.673i)T^{2} \)
37 \( 1 + 1.47T + T^{2} \)
41 \( 1 - 0.722iT - T^{2} \)
43 \( 1 + (0.739 + 0.673i)T^{2} \)
47 \( 1 + (-0.602 + 0.798i)T^{2} \)
53 \( 1 + (-0.576 + 1.48i)T + (-0.739 - 0.673i)T^{2} \)
59 \( 1 + (0.602 - 0.798i)T^{2} \)
61 \( 1 + (-0.876 + 1.75i)T + (-0.602 - 0.798i)T^{2} \)
67 \( 1 + (-0.850 + 0.526i)T^{2} \)
71 \( 1 + (-0.982 + 0.183i)T^{2} \)
73 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
79 \( 1 + (0.445 - 0.895i)T^{2} \)
83 \( 1 + (-0.273 + 0.961i)T^{2} \)
89 \( 1 + (-0.709 + 0.778i)T + (-0.0922 - 0.995i)T^{2} \)
97 \( 1 + (0.982 + 0.183i)T^{2} \)
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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

−11.06807113140726959552248432064, −10.29433172603554947215606790517, −9.067189388410742225962504953607, −8.452225667900535562102725981098, −6.82826683733714847544235633136, −6.16052917341035427235310661026, −5.19823661550487678781449221770, −3.68796143485761420384961449383, −3.34772458031369235805903892871, −1.51863454212355825346212035044, 2.35303804597685535450065165338, 3.76519005313831966346274282123, 4.62770342564273488987449155035, 5.69109694229175694721264944890, 6.50984586934345335894036660420, 7.56643189451348803541232562408, 8.469433162042317193065913293904, 8.984267584751938936694045234905, 10.55438594020013762597655409879, 11.45414895890251482526621481524

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