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Properties

Label 2-2667-2667.1250-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.955 - 0.295i$
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 + 3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + 7-s + i·8-s + 9-s + (−0.499 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + 3-s + (−0.866 + 0.5i)5-s + (0.866 − 0.5i)6-s + 7-s + i·8-s + 9-s + (−0.499 + 0.866i)10-s + (−0.866 + 0.5i)11-s + (−0.5 − 0.866i)13-s + (0.866 − 0.5i)14-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.955 - 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.385609559\)
\(L(\frac12)\) \(\approx\) \(2.385609559\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + T^{2} \)
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 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.999448908731240841057149308497, −7.87191803240359261411368015106, −7.77656186348105961029849153292, −7.29738339470056689867256107813, −5.42129174224092376608064451490, −5.11814067791228192419356217670, −3.99670722919855312756579440394, −3.46408876450260198054852781822, −2.71395439457617484092867800122, −1.78515156073823932457259628511, 1.19346832478074838323665877949, 2.54296427602262915853519846787, 3.68773673905682891327500914428, 4.23619086786042442375113863470, 5.04810042591923827515589794197, 5.55210890603272390416968628535, 6.98669774107580626675755058652, 7.44070469197094883407502905984, 8.095553647393264377309497668772, 8.914181481479986459584696217191

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