Properties

Label 2-548-548.115-c0-0-0
Degree $2$
Conductor $548$
Sign $-0.111 - 0.993i$
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  + (−0.602 + 0.798i)2-s + (−0.273 − 0.961i)4-s + (1.02 + 1.35i)5-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s − 1.70·10-s + (−0.404 + 0.368i)13-s + (−0.850 + 0.526i)16-s + (1.73 − 0.673i)17-s + (0.445 − 0.895i)18-s + (1.02 − 1.35i)20-s + (−0.517 + 1.81i)25-s + (−0.0505 − 0.544i)26-s + (−1.25 + 0.778i)29-s + (0.0922 − 0.995i)32-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (−0.273 − 0.961i)4-s + (1.02 + 1.35i)5-s + (0.932 + 0.361i)8-s + (−0.982 + 0.183i)9-s − 1.70·10-s + (−0.404 + 0.368i)13-s + (−0.850 + 0.526i)16-s + (1.73 − 0.673i)17-s + (0.445 − 0.895i)18-s + (1.02 − 1.35i)20-s + (−0.517 + 1.81i)25-s + (−0.0505 − 0.544i)26-s + (−1.25 + 0.778i)29-s + (0.0922 − 0.995i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $-0.111 - 0.993i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ -0.111 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7252781634\)
\(L(\frac12)\) \(\approx\) \(0.7252781634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.602 - 0.798i)T \)
137 \( 1 + (-0.739 - 0.673i)T \)
good3 \( 1 + (0.982 - 0.183i)T^{2} \)
5 \( 1 + (-1.02 - 1.35i)T + (-0.273 + 0.961i)T^{2} \)
7 \( 1 + (-0.0922 + 0.995i)T^{2} \)
11 \( 1 + (0.850 - 0.526i)T^{2} \)
13 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
17 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
19 \( 1 + (0.602 + 0.798i)T^{2} \)
23 \( 1 + (-0.445 + 0.895i)T^{2} \)
29 \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \)
31 \( 1 + (0.602 - 0.798i)T^{2} \)
37 \( 1 + 1.20T + T^{2} \)
41 \( 1 - 0.891T + T^{2} \)
43 \( 1 + (0.602 + 0.798i)T^{2} \)
47 \( 1 + (-0.932 + 0.361i)T^{2} \)
53 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
59 \( 1 + (-0.932 + 0.361i)T^{2} \)
61 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
67 \( 1 + (-0.0922 + 0.995i)T^{2} \)
71 \( 1 + (0.850 + 0.526i)T^{2} \)
73 \( 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2} \)
79 \( 1 + (0.982 + 0.183i)T^{2} \)
83 \( 1 + (-0.739 - 0.673i)T^{2} \)
89 \( 1 + (0.111 + 0.147i)T + (-0.273 + 0.961i)T^{2} \)
97 \( 1 + (0.547 + 1.92i)T + (-0.850 + 0.526i)T^{2} \)
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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

−10.95628083747396873497196750269, −10.13609728023994450741731298464, −9.594436530157889945728345814146, −8.608698873199444926930993561263, −7.43932179965714923686417738213, −6.88112345080707818419688562655, −5.76009662251857052763087476457, −5.32796444174973407008252315213, −3.29044307113393011269948855176, −2.04272797386058167027698342846, 1.20554678516377352515544647439, 2.52287918502321558569166031396, 3.87335129300715868365206847819, 5.25460328997336694463538784736, 5.87948144853072628234641485248, 7.60623619198971399679973609425, 8.406970967316681844089555539220, 9.153202442166451577870595261679, 9.800299694358263123463541833388, 10.57949809851774195666383545540

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