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  − 1.30·3-s − 3·5-s − 0.302·7-s − 1.30·9-s + 2.69·13-s + 3.90·15-s + 2.30·17-s + 2·19-s + 0.394·21-s − 2.30·23-s + 4·25-s + 5.60·27-s + 7.60·29-s + 6.60·31-s + 0.908·35-s + 0.394·37-s − 3.51·39-s − 6.21·41-s + 9.60·43-s + 3.90·45-s + 1.60·47-s − 6.90·49-s − 3·51-s − 4.60·53-s − 2.60·57-s + 7.81·59-s + 6.81·61-s + ⋯
L(s)  = 1  − 0.752·3-s − 1.34·5-s − 0.114·7-s − 0.434·9-s + 0.748·13-s + 1.00·15-s + 0.558·17-s + 0.458·19-s + 0.0860·21-s − 0.480·23-s + 0.800·25-s + 1.07·27-s + 1.41·29-s + 1.18·31-s + 0.153·35-s + 0.0648·37-s − 0.562·39-s − 0.970·41-s + 1.46·43-s + 0.582·45-s + 0.234·47-s − 0.986·49-s − 0.420·51-s − 0.632·53-s − 0.345·57-s + 1.01·59-s + 0.872·61-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

\(L(1)\) \(\approx\) \(0.784527\)
\(L(\frac12)\) \(\approx\) \(0.784527\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.30T + 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + 0.302T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 - 2.30T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 2.30T + 23T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 - 0.394T + 37T^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 - 9.60T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 + 4.60T + 53T^{2} \)
59 \( 1 - 7.81T + 59T^{2} \)
61 \( 1 - 6.81T + 61T^{2} \)
67 \( 1 - 8.21T + 67T^{2} \)
71 \( 1 - 3.90T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−10.73181878433097000391309121198, −9.808267262921383522661575283509, −8.476280579730050832206196218228, −8.054853667853943176173593229555, −6.90567599531242657940243573298, −6.05745921150348059150766290230, −5.02128472225635093282454747163, −3.99613816093496145148275393777, −2.98394981789951374605396405234, −0.78301635456452129501850509076, 0.78301635456452129501850509076, 2.98394981789951374605396405234, 3.99613816093496145148275393777, 5.02128472225635093282454747163, 6.05745921150348059150766290230, 6.90567599531242657940243573298, 8.054853667853943176173593229555, 8.476280579730050832206196218228, 9.808267262921383522661575283509, 10.73181878433097000391309121198

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