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Properties

Label 2-548-548.15-c0-0-0
Degree $2$
Conductor $548$
Sign $-0.517 + 0.855i$
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  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (−1.72 + 0.489i)5-s + (0.739 − 0.673i)8-s + (−0.932 − 0.361i)9-s − 1.79i·10-s + (−1.04 + 0.0971i)13-s + (0.445 + 0.895i)16-s + (−1.09 − 0.995i)17-s + (0.602 − 0.798i)18-s + (1.72 + 0.489i)20-s + (1.87 − 1.16i)25-s + (0.193 − 1.03i)26-s + (−1.78 + 0.887i)29-s + (−0.982 + 0.183i)32-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (−1.72 + 0.489i)5-s + (0.739 − 0.673i)8-s + (−0.932 − 0.361i)9-s − 1.79i·10-s + (−1.04 + 0.0971i)13-s + (0.445 + 0.895i)16-s + (−1.09 − 0.995i)17-s + (0.602 − 0.798i)18-s + (1.72 + 0.489i)20-s + (1.87 − 1.16i)25-s + (0.193 − 1.03i)26-s + (−1.78 + 0.887i)29-s + (−0.982 + 0.183i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ -0.517 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02529272141\)
\(L(\frac12)\) \(\approx\) \(0.02529272141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 - 0.961i)T \)
137 \( 1 + (-0.0922 + 0.995i)T \)
good3 \( 1 + (0.932 + 0.361i)T^{2} \)
5 \( 1 + (1.72 - 0.489i)T + (0.850 - 0.526i)T^{2} \)
7 \( 1 + (0.982 - 0.183i)T^{2} \)
11 \( 1 + (-0.445 - 0.895i)T^{2} \)
13 \( 1 + (1.04 - 0.0971i)T + (0.982 - 0.183i)T^{2} \)
17 \( 1 + (1.09 + 0.995i)T + (0.0922 + 0.995i)T^{2} \)
19 \( 1 + (0.273 + 0.961i)T^{2} \)
23 \( 1 + (-0.602 + 0.798i)T^{2} \)
29 \( 1 + (1.78 - 0.887i)T + (0.602 - 0.798i)T^{2} \)
31 \( 1 + (-0.273 + 0.961i)T^{2} \)
37 \( 1 - 0.547T + T^{2} \)
41 \( 1 - 1.59iT - T^{2} \)
43 \( 1 + (-0.273 - 0.961i)T^{2} \)
47 \( 1 + (0.739 + 0.673i)T^{2} \)
53 \( 1 + (-1.07 - 0.811i)T + (0.273 + 0.961i)T^{2} \)
59 \( 1 + (-0.739 - 0.673i)T^{2} \)
61 \( 1 + (0.831 - 0.322i)T + (0.739 - 0.673i)T^{2} \)
67 \( 1 + (-0.982 + 0.183i)T^{2} \)
71 \( 1 + (0.445 - 0.895i)T^{2} \)
73 \( 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (0.932 - 0.361i)T^{2} \)
83 \( 1 + (0.0922 - 0.995i)T^{2} \)
89 \( 1 + (0.353 - 0.100i)T + (0.850 - 0.526i)T^{2} \)
97 \( 1 + (-0.445 - 0.895i)T^{2} \)
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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

−11.45053348922643821187360566921, −10.86752734204758663056096856975, −9.506427977213081612307507930734, −8.800649515024331824825186306135, −7.81159205750336798776781976044, −7.28040150424329203138213164401, −6.42615392239082458953207844066, −5.08818903316950714268949017456, −4.19845279844451049745258330636, −3.00195683503452171713476523004, 0.03193937517844284572994644557, 2.29216518794228162780667763976, 3.63374822651492979724471709798, 4.37582451796074884318690532930, 5.41025905435446773859320400664, 7.22838306576750358673936252309, 8.054227624429116749846022056177, 8.594499137774893504784811924754, 9.508163264403989339730459173545, 10.78424674528053249675278931412

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