Properties

Label 2-660-1.1-c1-0-7
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s − 15-s − 6·17-s + 2·19-s − 4·21-s + 25-s + 27-s − 4·31-s − 33-s + 4·35-s − 10·37-s − 4·39-s − 4·43-s − 45-s + 12·47-s + 9·49-s − 6·51-s + 6·53-s + 55-s + 2·57-s + 12·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.458·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s − 0.640·39-s − 0.609·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.134·55-s + 0.264·57-s + 1.56·59-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: \(1\)
Selberg data: \((2,\ 660,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.978183703853620625557175039707, −9.238061894079419833722803309921, −8.510418078048314070850105018894, −7.22741549048532299546614518657, −6.87977465279522291645710913052, −5.54701550218547875334577656336, −4.30987323692753027270428495802, −3.30360101068721859971758002522, −2.35548221701648672137868113902, 0, 2.35548221701648672137868113902, 3.30360101068721859971758002522, 4.30987323692753027270428495802, 5.54701550218547875334577656336, 6.87977465279522291645710913052, 7.22741549048532299546614518657, 8.510418078048314070850105018894, 9.238061894079419833722803309921, 9.978183703853620625557175039707

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