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Properties

Label 2-663-663.662-c0-0-3
Degree $2$
Conductor $663$
Sign $1$
Analytic cond. $0.330880$
Root an. cond. $0.575221$
Motivic weight $0$
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 − 4-s + 9-s − 12-s + 13-s + 16-s + 17-s − 2·23-s + 25-s + 27-s − 2·29-s − 36-s + 39-s − 2·43-s + 48-s − 49-s + 51-s − 52-s − 64-s − 68-s − 2·69-s + 75-s + 81-s − 2·87-s + 2·92-s − 100-s + 2·103-s + ⋯
L(s)  = 1  + 3-s − 4-s + 9-s − 12-s + 13-s + 16-s + 17-s − 2·23-s + 25-s + 27-s − 2·29-s − 36-s + 39-s − 2·43-s + 48-s − 49-s + 51-s − 52-s − 64-s − 68-s − 2·69-s + 75-s + 81-s − 2·87-s + 2·92-s − 100-s + 2·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.330880\)
Root analytic conductor: \(0.575221\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{663} (662, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 663,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094193526\)
\(L(\frac12)\) \(\approx\) \(1.094193526\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
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

−10.36442542966915047767218568063, −9.773456334004245954421655108682, −8.945482226315416568707174254958, −8.230194535370850578268331177863, −7.59606865464983076191970161504, −6.23557832276311253784838868654, −5.13941353020728706272006432770, −3.95346256756245897133948543538, −3.35812742343703492628439424397, −1.63868089450478091222670068030, 1.63868089450478091222670068030, 3.35812742343703492628439424397, 3.95346256756245897133948543538, 5.13941353020728706272006432770, 6.23557832276311253784838868654, 7.59606865464983076191970161504, 8.230194535370850578268331177863, 8.945482226315416568707174254958, 9.773456334004245954421655108682, 10.36442542966915047767218568063

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