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Properties

Label 2-2667-2667.20-c0-0-1
Degree $2$
Conductor $2667$
Sign $0.949 - 0.314i$
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  + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (1.5 − 0.866i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + 25-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (−1.5 + 0.866i)31-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (1.5 − 0.866i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + 25-s − 0.999·27-s + (−0.5 − 0.866i)28-s + (−1.5 + 0.866i)31-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.818141406\)
\(L(\frac12)\) \(\approx\) \(1.818141406\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
127 \( 1 + T \)
good2 \( 1 - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 - 1.73iT - T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.918517471746351497190471725330, −8.523622301178345917558085633069, −7.42387908391387089342438516465, −6.94125375728224850543579157695, −6.00659488683973016505239534784, −5.17495144536049068046511151565, −4.13617256666272231453928785836, −3.25154917199644415961179196812, −2.81890893458273406718841028192, −1.29672646303066551903881978343, 1.53920944945359469686074550430, 2.10840939549587454275878092952, 3.25663883034370097249722724150, 3.77849897792914131633013719026, 5.55016567719090441599080121530, 6.14445385542595138834524093380, 6.59819472826760933640101174245, 7.48169972633327124760273299964, 8.177921974278925545278603175321, 8.906999054392651194462244992992

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