Properties

Label 2-2667-2667.107-c0-0-1
Degree $2$
Conductor $2667$
Sign $0.813 + 0.581i$
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  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.499i)6-s + (−0.5 + 0.866i)7-s i·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s + i·11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.499i)14-s + (0.866 − 0.499i)15-s + (0.5 − 0.866i)16-s i·17-s − 0.999i·18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.499i)6-s + (−0.5 + 0.866i)7-s i·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s + i·11-s + (−0.5 − 0.866i)13-s + (−0.866 + 0.499i)14-s + (0.866 − 0.499i)15-s + (0.5 − 0.866i)16-s i·17-s − 0.999i·18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8737431936\)
\(L(\frac12)\) \(\approx\) \(0.8737431936\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.163554470528310815585079771380, −8.154914227244949135835321941430, −7.19439360972823559535037813626, −6.51978781312262476474856390335, −5.45970309683497098419536603793, −5.12297451356591047288360220366, −4.47320360553104602686972541932, −3.62021712429457352504773802963, −2.72278983682119099063468410852, −0.48135976976241360602414322786, 1.37720298372323427136629263249, 2.79297315653459468345230347091, 3.41758204157106283801856732589, 4.27902557138296174432113779386, 5.00568009667819703071630147223, 6.31261346320391692740356884802, 6.49918032043493680527505861284, 7.72662212100072625809239269001, 7.990129098618448704167724439144, 8.901834095451085869614143988875

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