Properties

Degree $2$
Conductor $668$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 1.89·3-s + 2·5-s + 4.92·7-s + 0.601·9-s − 2.15·11-s − 3.32·13-s + 3.79·15-s + 5.91·17-s − 0.927·19-s + 9.33·21-s − 7.63·23-s − 25-s − 4.55·27-s − 1.59·29-s − 5.18·31-s − 4.09·33-s + 9.84·35-s + 1.53·37-s − 6.31·39-s − 9.71·41-s + 6.65·43-s + 1.20·45-s + 5.72·47-s + 17.2·49-s + 11.2·51-s − 9.24·53-s − 4.31·55-s + ⋯
L(s)  = 1  + 1.09·3-s + 0.894·5-s + 1.85·7-s + 0.200·9-s − 0.649·11-s − 0.922·13-s + 0.979·15-s + 1.43·17-s − 0.212·19-s + 2.03·21-s − 1.59·23-s − 0.200·25-s − 0.876·27-s − 0.296·29-s − 0.930·31-s − 0.712·33-s + 1.66·35-s + 0.251·37-s − 1.01·39-s − 1.51·41-s + 1.01·43-s + 0.179·45-s + 0.834·47-s + 2.45·49-s + 1.57·51-s − 1.26·53-s − 0.581·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{668} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66379\)
\(L(\frac12)\) \(\approx\) \(2.66379\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 + T \)
good3 \( 1 - 1.89T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 4.92T + 7T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 + 0.927T + 19T^{2} \)
23 \( 1 + 7.63T + 23T^{2} \)
29 \( 1 + 1.59T + 29T^{2} \)
31 \( 1 + 5.18T + 31T^{2} \)
37 \( 1 - 1.53T + 37T^{2} \)
41 \( 1 + 9.71T + 41T^{2} \)
43 \( 1 - 6.65T + 43T^{2} \)
47 \( 1 - 5.72T + 47T^{2} \)
53 \( 1 + 9.24T + 53T^{2} \)
59 \( 1 - 5.78T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 8.23T + 79T^{2} \)
83 \( 1 - 7.97T + 83T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.25343884565562589118459553320, −9.700101264486565439160150642127, −8.625998352595403000151378837362, −7.938045256444632597035754417979, −7.46327189081337935861031128684, −5.72192644241367696862147939740, −5.15780096951604515523804045145, −3.87689014113227537308151753362, −2.42838613076373027708445734664, −1.79058045829787187667008636688, 1.79058045829787187667008636688, 2.42838613076373027708445734664, 3.87689014113227537308151753362, 5.15780096951604515523804045145, 5.72192644241367696862147939740, 7.46327189081337935861031128684, 7.938045256444632597035754417979, 8.625998352595403000151378837362, 9.700101264486565439160150642127, 10.25343884565562589118459553320

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