Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.12·3-s + 2·5-s − 0.516·7-s + 6.77·9-s + 0.538·11-s − 0.295·13-s + 6.25·15-s − 6.99·17-s − 4.06·19-s − 1.61·21-s + 0.781·23-s − 25-s + 11.7·27-s + 0.811·29-s + 3.07·31-s + 1.68·33-s − 1.03·35-s − 3.95·37-s − 0.923·39-s + 0.743·41-s + 0.590·43-s + 13.5·45-s − 3.47·47-s − 6.73·49-s − 21.8·51-s + 6.69·53-s + 1.07·55-s + ⋯
L(s)  = 1  + 1.80·3-s + 0.894·5-s − 0.195·7-s + 2.25·9-s + 0.162·11-s − 0.0819·13-s + 1.61·15-s − 1.69·17-s − 0.932·19-s − 0.352·21-s + 0.162·23-s − 0.200·25-s + 2.26·27-s + 0.150·29-s + 0.552·31-s + 0.292·33-s − 0.174·35-s − 0.650·37-s − 0.147·39-s + 0.116·41-s + 0.0900·43-s + 2.01·45-s − 0.506·47-s − 0.961·49-s − 3.06·51-s + 0.920·53-s + 0.145·55-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

\(L(1)\) \(\approx\) \(2.96207\)
\(L(\frac12)\) \(\approx\) \(2.96207\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.12T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 0.516T + 7T^{2} \)
11 \( 1 - 0.538T + 11T^{2} \)
13 \( 1 + 0.295T + 13T^{2} \)
17 \( 1 + 6.99T + 17T^{2} \)
19 \( 1 + 4.06T + 19T^{2} \)
23 \( 1 - 0.781T + 23T^{2} \)
29 \( 1 - 0.811T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 - 0.743T + 41T^{2} \)
43 \( 1 - 0.590T + 43T^{2} \)
47 \( 1 + 3.47T + 47T^{2} \)
53 \( 1 - 6.69T + 53T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 - 4.63T + 61T^{2} \)
67 \( 1 - 6.55T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 4.88T + 79T^{2} \)
83 \( 1 + 1.11T + 83T^{2} \)
89 \( 1 + 1.79T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−10.14715990398771958684751913622, −9.507663347510424358394650225231, −8.793013581347902825245043039905, −8.207126025764016844185651028517, −7.02928972045614198766726718070, −6.31133820922531379592352314443, −4.74384023820616629854639779279, −3.76603271965913903382684735729, −2.56223257888144330032574909925, −1.87517241733764295673242023254, 1.87517241733764295673242023254, 2.56223257888144330032574909925, 3.76603271965913903382684735729, 4.74384023820616629854639779279, 6.31133820922531379592352314443, 7.02928972045614198766726718070, 8.207126025764016844185651028517, 8.793013581347902825245043039905, 9.507663347510424358394650225231, 10.14715990398771958684751913622

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