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  − 1.83·3-s + 2·5-s + 1.53·7-s + 0.363·9-s + 6.05·11-s − 5.50·13-s − 3.66·15-s + 1.11·17-s − 2.87·19-s − 2.81·21-s + 6.59·23-s − 25-s + 4.83·27-s + 3.97·29-s − 1.23·31-s − 11.0·33-s + 3.07·35-s + 11.1·37-s + 10.1·39-s + 2.55·41-s + 11.0·43-s + 0.726·45-s + 8.14·47-s − 4.63·49-s − 2.04·51-s − 6.62·53-s + 12.1·55-s + ⋯
L(s)  = 1  − 1.05·3-s + 0.894·5-s + 0.580·7-s + 0.121·9-s + 1.82·11-s − 1.52·13-s − 0.946·15-s + 0.270·17-s − 0.658·19-s − 0.614·21-s + 1.37·23-s − 0.200·25-s + 0.930·27-s + 0.737·29-s − 0.221·31-s − 1.93·33-s + 0.519·35-s + 1.83·37-s + 1.61·39-s + 0.398·41-s + 1.68·43-s + 0.108·45-s + 1.18·47-s − 0.662·49-s − 0.286·51-s − 0.910·53-s + 1.63·55-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

\(L(1)\) \(\approx\) \(1.33246\)
\(L(\frac12)\) \(\approx\) \(1.33246\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.83T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 - 6.05T + 11T^{2} \)
13 \( 1 + 5.50T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 + 2.87T + 19T^{2} \)
23 \( 1 - 6.59T + 23T^{2} \)
29 \( 1 - 3.97T + 29T^{2} \)
31 \( 1 + 1.23T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 8.14T + 47T^{2} \)
53 \( 1 + 6.62T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 0.918T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 5.47T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 4.26T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−10.70543134669557328655652853184, −9.547900290096318874526848219177, −9.144504089377188658645776407362, −7.75814694008679274060591013996, −6.66990870405770489016782227139, −6.07783765845692870716981724547, −5.10908223471499265317748868139, −4.31478346351207238068220125386, −2.53370254557324170740912158352, −1.12079404176756742640723923648, 1.12079404176756742640723923648, 2.53370254557324170740912158352, 4.31478346351207238068220125386, 5.10908223471499265317748868139, 6.07783765845692870716981724547, 6.66990870405770489016782227139, 7.75814694008679274060591013996, 9.144504089377188658645776407362, 9.547900290096318874526848219177, 10.70543134669557328655652853184

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