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Properties

Label 2-548-548.123-c0-0-0
Degree $2$
Conductor $548$
Sign $0.809 - 0.587i$
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 + (1.02 + 0.634i)5-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s − 1.20·10-s + (−0.876 − 0.163i)13-s + (−0.602 − 0.798i)16-s + (0.0170 − 0.183i)17-s + (−0.273 + 0.961i)18-s + (1.02 − 0.634i)20-s + (0.201 + 0.405i)25-s + (0.831 − 0.322i)26-s + (1.18 + 1.56i)29-s + (0.932 + 0.361i)32-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.445 − 0.895i)4-s + (1.02 + 0.634i)5-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s − 1.20·10-s + (−0.876 − 0.163i)13-s + (−0.602 − 0.798i)16-s + (0.0170 − 0.183i)17-s + (−0.273 + 0.961i)18-s + (1.02 − 0.634i)20-s + (0.201 + 0.405i)25-s + (0.831 − 0.322i)26-s + (1.18 + 1.56i)29-s + (0.932 + 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.809 - 0.587i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7352308366\)
\(L(\frac12)\) \(\approx\) \(0.7352308366\)
\(L(1)\) 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 \( 1 + (-1.02 - 0.634i)T + (0.445 + 0.895i)T^{2} \)
7 \( 1 + (-0.932 - 0.361i)T^{2} \)
11 \( 1 + (0.602 + 0.798i)T^{2} \)
13 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
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 \( 1 + (-1.18 - 1.56i)T + (-0.273 + 0.961i)T^{2} \)
31 \( 1 + (0.850 - 0.526i)T^{2} \)
37 \( 1 + 1.70T + T^{2} \)
41 \( 1 + 0.547T + T^{2} \)
43 \( 1 + (0.850 + 0.526i)T^{2} \)
47 \( 1 + (-0.0922 + 0.995i)T^{2} \)
53 \( 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2} \)
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 \( 1 + (1.58 + 0.981i)T + (0.445 + 0.895i)T^{2} \)
97 \( 1 + (-0.891 + 1.79i)T + (-0.602 - 0.798i)T^{2} \)
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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

−10.53320300920943658508933113790, −10.20426696055356966327405039790, −9.403492638037084238136529352574, −8.597147424214292289997781631968, −7.24754791316507779029579761123, −6.80966951473213204639032514337, −5.85328483367670936756980709188, −4.82246872082922944008096649610, −2.95748019265929001719007852377, −1.64088918928215887831913089905, 1.56660473720975213498296493320, 2.53606125521311599423913339256, 4.21687931675048143212310111655, 5.27932026831383470651153221121, 6.57807231473082416729138127166, 7.52366693578752657067105239073, 8.454069611312507638214846407596, 9.332846137530892953969422826019, 10.07250032434451098147870220013, 10.51813349439605944122039818770

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