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  + 2.30·3-s − 3·5-s + 3.30·7-s + 2.30·9-s + 6.30·13-s − 6.90·15-s − 1.30·17-s + 2·19-s + 7.60·21-s + 1.30·23-s + 4·25-s − 1.60·27-s + 0.394·29-s − 0.605·31-s − 9.90·35-s + 7.60·37-s + 14.5·39-s + 8.21·41-s + 2.39·43-s − 6.90·45-s − 5.60·47-s + 3.90·49-s − 3·51-s + 2.60·53-s + 4.60·57-s − 13.8·59-s − 14.8·61-s + ⋯
L(s)  = 1  + 1.32·3-s − 1.34·5-s + 1.24·7-s + 0.767·9-s + 1.74·13-s − 1.78·15-s − 0.315·17-s + 0.458·19-s + 1.65·21-s + 0.271·23-s + 0.800·25-s − 0.308·27-s + 0.0732·29-s − 0.108·31-s − 1.67·35-s + 1.25·37-s + 2.32·39-s + 1.28·41-s + 0.365·43-s − 1.02·45-s − 0.817·47-s + 0.558·49-s − 0.420·51-s + 0.357·53-s + 0.610·57-s − 1.79·59-s − 1.89·61-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

\(L(1)\) \(\approx\) \(2.20443\)
\(L(\frac12)\) \(\approx\) \(2.20443\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30T + 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 + 1.30T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 1.30T + 23T^{2} \)
29 \( 1 - 0.394T + 29T^{2} \)
31 \( 1 + 0.605T + 31T^{2} \)
37 \( 1 - 7.60T + 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 - 2.39T + 43T^{2} \)
47 \( 1 + 5.60T + 47T^{2} \)
53 \( 1 - 2.60T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 6.21T + 67T^{2} \)
71 \( 1 + 6.90T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 - 7.81T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.78684176606250352780348090690, −9.220792238273381390054638834383, −8.671315156529340627781238131251, −7.82032309207390708230357363588, −7.65098058816185145766008554081, −6.10159033130350986165324583392, −4.59745577066120010943531232006, −3.87332638670442420979740841884, −2.94107401849814219392407337671, −1.41901172003878071098671570951, 1.41901172003878071098671570951, 2.94107401849814219392407337671, 3.87332638670442420979740841884, 4.59745577066120010943531232006, 6.10159033130350986165324583392, 7.65098058816185145766008554081, 7.82032309207390708230357363588, 8.671315156529340627781238131251, 9.220792238273381390054638834383, 10.78684176606250352780348090690

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