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Properties

Label 2-548-548.547-c0-0-1
Degree $2$
Conductor $548$
Sign $1$
Analytic cond. $0.273487$
Root an. cond. $0.522960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.00·18-s − 1.41·23-s − 1.41·24-s + 25-s − 1.41·31-s − 32-s + 1.00·36-s − 1.41·43-s + 1.41·46-s + 1.41·47-s + 1.41·48-s + 49-s − 50-s + 1.41·62-s + 64-s + 1.41·67-s − 2.00·69-s − 1.41·71-s − 1.00·72-s + ⋯
L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·6-s − 8-s + 1.00·9-s + 1.41·12-s + 16-s − 1.00·18-s − 1.41·23-s − 1.41·24-s + 25-s − 1.41·31-s − 32-s + 1.00·36-s − 1.41·43-s + 1.41·46-s + 1.41·47-s + 1.41·48-s + 49-s − 50-s + 1.41·62-s + 64-s + 1.41·67-s − 2.00·69-s − 1.41·71-s − 1.00·72-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $1$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (547, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8968955316\)
\(L(\frac12)\) \(\approx\) \(0.8968955316\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
137 \( 1 - T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.67756105353010294689138041150, −9.910359053643926708417676060003, −9.068468491916356284943757047961, −8.510298013201381740630429050438, −7.70040086457106388972850218228, −6.93656090116850468828156912594, −5.67823058814905872449830864793, −3.92498967789033671437052662418, −2.86504915259262455866205921364, −1.82857590280282073748805575419, 1.82857590280282073748805575419, 2.86504915259262455866205921364, 3.92498967789033671437052662418, 5.67823058814905872449830864793, 6.93656090116850468828156912594, 7.70040086457106388972850218228, 8.510298013201381740630429050438, 9.068468491916356284943757047961, 9.910359053643926708417676060003, 10.67756105353010294689138041150

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