Properties

Label 2-660-1.1-c1-0-0
Degree $2$
Conductor $660$
Sign $1$
Analytic cond. $5.27012$
Root an. cond. $2.29567$
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  − 3-s − 5-s − 2·7-s + 9-s − 11-s + 2·13-s + 15-s + 8·17-s − 2·19-s + 2·21-s + 8·23-s + 25-s − 27-s + 33-s + 2·35-s + 2·37-s − 2·39-s + 6·43-s − 45-s + 8·47-s − 3·49-s − 8·51-s + 6·53-s + 55-s + 2·57-s − 4·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.94·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.174·33-s + 0.338·35-s + 0.328·37-s − 0.320·39-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 1.12·51-s + 0.824·53-s + 0.134·55-s + 0.264·57-s − 0.520·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(5.27012\)
Root analytic conductor: \(2.29567\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{660} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 660,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070023212\)
\(L(\frac12)\) \(\approx\) \(1.070023212\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−10.58917076131701257357177911550, −9.788620372840369960521330671071, −8.850361822859748464532686119818, −7.78412586857067234094953487514, −6.98759403795642026924393450017, −5.99158381383684602539992031128, −5.17810728666522882356486400345, −3.91288399319096089362633398104, −2.94270272449041975847268775885, −0.938183538879467721656887858535, 0.938183538879467721656887858535, 2.94270272449041975847268775885, 3.91288399319096089362633398104, 5.17810728666522882356486400345, 5.99158381383684602539992031128, 6.98759403795642026924393450017, 7.78412586857067234094953487514, 8.850361822859748464532686119818, 9.788620372840369960521330671071, 10.58917076131701257357177911550

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