Properties

Degree $2$
Conductor $204$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s + 17-s − 19-s − 23-s + 27-s + 2·29-s − 33-s − 39-s − 41-s − 43-s − 45-s + 49-s + 51-s + 55-s − 57-s + 65-s + 2·67-s − 69-s + 2·71-s + 81-s − 85-s + 2·87-s + ⋯
L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 13-s − 15-s + 17-s − 19-s − 23-s + 27-s + 2·29-s − 33-s − 39-s − 41-s − 43-s − 45-s + 49-s + 51-s + 55-s − 57-s + 65-s + 2·67-s − 69-s + 2·71-s + 81-s − 85-s + 2·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(204\)    =    \(2^{2} \cdot 3 \cdot 17\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{204} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 204,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7863288172\)
\(L(\frac12)\) \(\approx\) \(0.7863288172\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−12.55293235020929802988028909026, −11.97716062598023379286844566197, −10.47631458520706009380759975999, −9.805222121350042209772523186414, −8.284936952968746021179779787637, −7.965202984910309898471426703055, −6.82424899172616989471966838318, −4.97602311316391520571027185228, −3.79943974802985091918625662732, −2.51256673938332341097950717807, 2.51256673938332341097950717807, 3.79943974802985091918625662732, 4.97602311316391520571027185228, 6.82424899172616989471966838318, 7.965202984910309898471426703055, 8.284936952968746021179779787637, 9.805222121350042209772523186414, 10.47631458520706009380759975999, 11.97716062598023379286844566197, 12.55293235020929802988028909026

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