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Properties

Label 2-2667-2667.1280-c0-0-1
Degree $2$
Conductor $2667$
Sign $0.0256 + 0.999i$
Analytic cond. $1.33100$
Root an. cond. $1.15369$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.733 − 0.680i)3-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)7-s + (0.0747 − 0.997i)9-s + (−0.0747 − 0.997i)12-s + (−1.26 − 0.496i)13-s + (−0.222 − 0.974i)16-s + 1.12i·19-s + (0.5 − 0.866i)21-s + (−0.222 + 0.974i)25-s + (−0.623 − 0.781i)27-s + (0.365 − 0.930i)28-s + (−0.0878 + 0.582i)31-s + (−0.733 − 0.680i)36-s + (0.365 − 0.632i)37-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)3-s + (0.623 − 0.781i)4-s + (0.955 − 0.294i)7-s + (0.0747 − 0.997i)9-s + (−0.0747 − 0.997i)12-s + (−1.26 − 0.496i)13-s + (−0.222 − 0.974i)16-s + 1.12i·19-s + (0.5 − 0.866i)21-s + (−0.222 + 0.974i)25-s + (−0.623 − 0.781i)27-s + (0.365 − 0.930i)28-s + (−0.0878 + 0.582i)31-s + (−0.733 − 0.680i)36-s + (0.365 − 0.632i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0256 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0256 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.0256 + 0.999i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.0256 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.881579950\)
\(L(\frac12)\) \(\approx\) \(1.881579950\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.955 + 0.294i)T \)
127 \( 1 + T \)
good2 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (1.26 + 0.496i)T + (0.733 + 0.680i)T^{2} \)
17 \( 1 + (0.955 - 0.294i)T^{2} \)
19 \( 1 - 1.12iT - T^{2} \)
23 \( 1 + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (0.826 + 0.563i)T^{2} \)
31 \( 1 + (0.0878 - 0.582i)T + (-0.955 - 0.294i)T^{2} \)
37 \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.955 - 0.294i)T^{2} \)
43 \( 1 + (0.255 - 0.829i)T + (-0.826 - 0.563i)T^{2} \)
47 \( 1 + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.290 + 0.0663i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.17 - 1.26i)T + (-0.0747 + 0.997i)T^{2} \)
71 \( 1 + (-0.365 - 0.930i)T^{2} \)
73 \( 1 + (0.865 - 1.79i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.826 - 0.563i)T + (0.365 + 0.930i)T^{2} \)
83 \( 1 + (0.988 + 0.149i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.425 - 0.131i)T + (0.826 - 0.563i)T^{2} \)
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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

−8.733342954552694206084014109607, −7.88623885277979290677180487844, −7.41904983196230320517867186565, −6.77123842848309692444768105137, −5.74219916279365445325646584080, −5.12092949051211771632764924324, −3.98794470260595781744854699864, −2.84301028446457482707004940139, −1.99826920222987552045849723269, −1.18027563392145039236645191795, 2.08427281121610655966481328094, 2.48745138942945673390404428620, 3.54377240939779611721926932343, 4.55572314483789000164519230668, 4.95047543438704689876755709563, 6.25010309635056009423381792432, 7.26828161981865735235328770577, 7.72519817088414625842745647851, 8.479504397627151458305885535083, 9.061102620892626877112227576887

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