Properties

Degree $2$
Conductor $2667$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15·2-s + 3-s + 2.64·4-s − 0.239·5-s − 2.15·6-s + 7-s − 1.40·8-s + 9-s + 0.515·10-s − 3.99·11-s + 2.64·12-s − 1.61·13-s − 2.15·14-s − 0.239·15-s − 2.27·16-s + 4.61·17-s − 2.15·18-s − 4.17·19-s − 0.633·20-s + 21-s + 8.62·22-s + 5.22·23-s − 1.40·24-s − 4.94·25-s + 3.48·26-s + 27-s + 2.64·28-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.577·3-s + 1.32·4-s − 0.106·5-s − 0.880·6-s + 0.377·7-s − 0.495·8-s + 0.333·9-s + 0.163·10-s − 1.20·11-s + 0.764·12-s − 0.448·13-s − 0.576·14-s − 0.0617·15-s − 0.569·16-s + 1.11·17-s − 0.508·18-s − 0.957·19-s − 0.141·20-s + 0.218·21-s + 1.83·22-s + 1.08·23-s − 0.285·24-s − 0.988·25-s + 0.683·26-s + 0.192·27-s + 0.500·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2667} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2667,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 + 2.15T + 2T^{2} \)
5 \( 1 + 0.239T + 5T^{2} \)
11 \( 1 + 3.99T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 5.22T + 23T^{2} \)
29 \( 1 - 9.10T + 29T^{2} \)
31 \( 1 + 6.57T + 31T^{2} \)
37 \( 1 + 0.353T + 37T^{2} \)
41 \( 1 + 7.72T + 41T^{2} \)
43 \( 1 + 7.33T + 43T^{2} \)
47 \( 1 - 3.52T + 47T^{2} \)
53 \( 1 + 0.655T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + 7.05T + 61T^{2} \)
67 \( 1 + 0.167T + 67T^{2} \)
71 \( 1 - 6.98T + 71T^{2} \)
73 \( 1 + 2.79T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 6.75T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 1.79T + 97T^{2} \)
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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

−8.366341346804682346408885673434, −7.977685499377859733412580498619, −7.36761416680151942248353610019, −6.58902676554451421343333300555, −5.34825248132872967815931881445, −4.56100039758419152895686563053, −3.23898604671392589707376902050, −2.35067960140450992167580650730, −1.40355511455004245193478482159, 0, 1.40355511455004245193478482159, 2.35067960140450992167580650730, 3.23898604671392589707376902050, 4.56100039758419152895686563053, 5.34825248132872967815931881445, 6.58902676554451421343333300555, 7.36761416680151942248353610019, 7.977685499377859733412580498619, 8.366341346804682346408885673434

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