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Properties

Label 2-2163-2163.1175-c0-0-0
Degree $2$
Conductor $2163$
Sign $-0.767 + 0.640i$
Analytic cond. $1.07947$
Root an. cond. $1.03897$
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  + (−0.739 + 0.673i)3-s + (−0.982 − 0.183i)4-s + (−0.445 + 0.895i)7-s + (0.0922 − 0.995i)9-s + (0.850 − 0.526i)12-s + (0.353 − 0.100i)13-s + (0.932 + 0.361i)16-s + (−1.20 + 1.32i)19-s + (−0.273 − 0.961i)21-s + (−0.602 + 0.798i)25-s + (0.602 + 0.798i)27-s + (0.602 − 0.798i)28-s + (−1.58 − 0.614i)31-s + (−0.273 + 0.961i)36-s + (−1.01 − 1.63i)37-s + ⋯
L(s)  = 1  + (−0.739 + 0.673i)3-s + (−0.982 − 0.183i)4-s + (−0.445 + 0.895i)7-s + (0.0922 − 0.995i)9-s + (0.850 − 0.526i)12-s + (0.353 − 0.100i)13-s + (0.932 + 0.361i)16-s + (−1.20 + 1.32i)19-s + (−0.273 − 0.961i)21-s + (−0.602 + 0.798i)25-s + (0.602 + 0.798i)27-s + (0.602 − 0.798i)28-s + (−1.58 − 0.614i)31-s + (−0.273 + 0.961i)36-s + (−1.01 − 1.63i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2163\)    =    \(3 \cdot 7 \cdot 103\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(1.07947\)
Root analytic conductor: \(1.03897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2163} (1175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2163,\ (\ :0),\ -0.767 + 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03870933523\)
\(L(\frac12)\) \(\approx\) \(0.03870933523\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (0.445 - 0.895i)T \)
103 \( 1 + (0.850 - 0.526i)T \)
good2 \( 1 + (0.982 + 0.183i)T^{2} \)
5 \( 1 + (0.602 - 0.798i)T^{2} \)
11 \( 1 + (-0.982 - 0.183i)T^{2} \)
13 \( 1 + (-0.353 + 0.100i)T + (0.850 - 0.526i)T^{2} \)
17 \( 1 + (-0.273 - 0.961i)T^{2} \)
19 \( 1 + (1.20 - 1.32i)T + (-0.0922 - 0.995i)T^{2} \)
23 \( 1 + (0.982 - 0.183i)T^{2} \)
29 \( 1 + (0.602 - 0.798i)T^{2} \)
31 \( 1 + (1.58 + 0.614i)T + (0.739 + 0.673i)T^{2} \)
37 \( 1 + (1.01 + 1.63i)T + (-0.445 + 0.895i)T^{2} \)
41 \( 1 + (-0.602 - 0.798i)T^{2} \)
43 \( 1 + (-0.840 + 1.35i)T + (-0.445 - 0.895i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.0922 + 0.995i)T^{2} \)
59 \( 1 + (-0.850 - 0.526i)T^{2} \)
61 \( 1 + (0.576 + 0.435i)T + (0.273 + 0.961i)T^{2} \)
67 \( 1 + (1.91 + 0.544i)T + (0.850 + 0.526i)T^{2} \)
71 \( 1 + (-0.602 - 0.798i)T^{2} \)
73 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
79 \( 1 + (-0.243 - 0.489i)T + (-0.602 + 0.798i)T^{2} \)
83 \( 1 + (-0.850 + 0.526i)T^{2} \)
89 \( 1 + (-0.932 + 0.361i)T^{2} \)
97 \( 1 + (1.27 - 0.961i)T + (0.273 - 0.961i)T^{2} \)
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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

−9.515644028210563423374121351970, −9.179128127235080566502122622372, −8.489222152277331196791312265236, −7.44002701399252249448086622105, −6.16645654850863798054957802055, −5.74331145313996715856223704156, −5.12098376904354333996997658967, −3.94464086937064145339122686727, −3.59445975244613063137715580068, −1.87579103999083795585170534666, 0.03201471120419475017162900539, 1.38630907840325551799705102921, 2.90937339736564216089744140204, 4.12500135896152163601786773122, 4.66522834283735017518281460046, 5.64873372742608297625410016064, 6.53215979252415698780479957593, 7.13153258193260097501135524003, 8.006263212966426359232971272398, 8.702910058532792194847263834217

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