Properties

Label 2-272-1.1-c1-0-4
Degree $2$
Conductor $272$
Sign $1$
Analytic cond. $2.17193$
Root an. cond. $1.47374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s + 9-s − 6·11-s + 2·13-s − 17-s + 4·19-s + 8·21-s − 5·25-s − 4·27-s + 4·31-s − 12·33-s − 4·37-s + 4·39-s + 6·41-s − 8·43-s + 9·49-s − 2·51-s − 6·53-s + 8·57-s − 4·61-s + 4·63-s − 8·67-s + 2·73-s − 10·75-s − 24·77-s − 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s + 1.74·21-s − 25-s − 0.769·27-s + 0.718·31-s − 2.08·33-s − 0.657·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 1.05·57-s − 0.512·61-s + 0.503·63-s − 0.977·67-s + 0.234·73-s − 1.15·75-s − 2.73·77-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $1$
Analytic conductor: \(2.17193\)
Root analytic conductor: \(1.47374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.864175057\)
\(L(\frac12)\) \(\approx\) \(1.864175057\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−11.78868723809074966331144526462, −10.96199091064046403542442582980, −9.983024040967094825270124003185, −8.763699502340737285319959905301, −8.020764485917199688918114818768, −7.56004329429510605274038490355, −5.66320470137327715389850473544, −4.65035915337384466130876980903, −3.14432806107716120656228384149, −1.95520344137895444953340586126, 1.95520344137895444953340586126, 3.14432806107716120656228384149, 4.65035915337384466130876980903, 5.66320470137327715389850473544, 7.56004329429510605274038490355, 8.020764485917199688918114818768, 8.763699502340737285319959905301, 9.983024040967094825270124003185, 10.96199091064046403542442582980, 11.78868723809074966331144526462

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