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  − 1.24·2-s + 3-s − 0.455·4-s − 3.33·5-s − 1.24·6-s + 7-s + 3.05·8-s + 9-s + 4.14·10-s + 2.90·11-s − 0.455·12-s − 4.59·13-s − 1.24·14-s − 3.33·15-s − 2.88·16-s − 5.29·17-s − 1.24·18-s + 1.86·19-s + 1.51·20-s + 21-s − 3.61·22-s + 5.03·23-s + 3.05·24-s + 6.11·25-s + 5.70·26-s + 27-s − 0.455·28-s + ⋯
L(s)  = 1  − 0.878·2-s + 0.577·3-s − 0.227·4-s − 1.49·5-s − 0.507·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s + 1.31·10-s + 0.876·11-s − 0.131·12-s − 1.27·13-s − 0.332·14-s − 0.860·15-s − 0.720·16-s − 1.28·17-s − 0.292·18-s + 0.428·19-s + 0.339·20-s + 0.218·21-s − 0.770·22-s + 1.04·23-s + 0.622·24-s + 1.22·25-s + 1.11·26-s + 0.192·27-s − 0.0860·28-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 \( 1 + 1.24T + 2T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
11 \( 1 - 2.90T + 11T^{2} \)
13 \( 1 + 4.59T + 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 1.86T + 19T^{2} \)
23 \( 1 - 5.03T + 23T^{2} \)
29 \( 1 - 3.69T + 29T^{2} \)
31 \( 1 - 0.968T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 - 2.54T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 0.666T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 - 9.88T + 71T^{2} \)
73 \( 1 + 2.57T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 0.493T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−8.580413127774544454982163327863, −7.71111113506960528873433446527, −7.40482016706651980532329718825, −6.63010885680894840288420615398, −4.92154005691844983084830983235, −4.48019072528173150371199271021, −3.70940389272013307579683991354, −2.57028550379554700689755255949, −1.24025408756021894088158879940, 0, 1.24025408756021894088158879940, 2.57028550379554700689755255949, 3.70940389272013307579683991354, 4.48019072528173150371199271021, 4.92154005691844983084830983235, 6.63010885680894840288420615398, 7.40482016706651980532329718825, 7.71111113506960528873433446527, 8.580413127774544454982163327863

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