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  + 0.246·2-s + 3-s − 1.93·4-s − 0.318·5-s + 0.246·6-s + 7-s − 0.969·8-s + 9-s − 0.0783·10-s + 2.79·11-s − 1.93·12-s − 6.49·13-s + 0.246·14-s − 0.318·15-s + 3.63·16-s − 1.33·17-s + 0.246·18-s + 4.36·19-s + 0.617·20-s + 21-s + 0.689·22-s − 5.29·23-s − 0.969·24-s − 4.89·25-s − 1.60·26-s + 27-s − 1.93·28-s + ⋯
L(s)  = 1  + 0.174·2-s + 0.577·3-s − 0.969·4-s − 0.142·5-s + 0.100·6-s + 0.377·7-s − 0.342·8-s + 0.333·9-s − 0.0247·10-s + 0.844·11-s − 0.559·12-s − 1.80·13-s + 0.0658·14-s − 0.0821·15-s + 0.909·16-s − 0.322·17-s + 0.0580·18-s + 1.00·19-s + 0.137·20-s + 0.218·21-s + 0.146·22-s − 1.10·23-s − 0.197·24-s − 0.979·25-s − 0.313·26-s + 0.192·27-s − 0.366·28-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 \( 1 - 0.246T + 2T^{2} \)
5 \( 1 + 0.318T + 5T^{2} \)
11 \( 1 - 2.79T + 11T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + 4.76T + 31T^{2} \)
37 \( 1 - 1.06T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 - 8.04T + 43T^{2} \)
47 \( 1 + 7.16T + 47T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
59 \( 1 + 5.02T + 59T^{2} \)
61 \( 1 + 2.48T + 61T^{2} \)
67 \( 1 - 0.549T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 1.49T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 7.20T + 97T^{2} \)
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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

−8.546921341492330181885799681287, −7.63934513967805420733838942281, −7.30949400107722559280277794780, −5.99768189424270587415592405735, −5.20976027816489578293496842880, −4.36664316523829153446406684357, −3.82083415866875313715198795380, −2.73082487966252966850486220613, −1.59943812692071389512082881175, 0, 1.59943812692071389512082881175, 2.73082487966252966850486220613, 3.82083415866875313715198795380, 4.36664316523829153446406684357, 5.20976027816489578293496842880, 5.99768189424270587415592405735, 7.30949400107722559280277794780, 7.63934513967805420733838942281, 8.546921341492330181885799681287

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