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Properties

Label 2-2163-2163.713-c0-0-1
Degree $2$
Conductor $2163$
Sign $-0.226 + 0.973i$
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  + (−0.0922 − 0.995i)3-s + (0.932 − 0.361i)4-s + (0.602 − 0.798i)7-s + (−0.982 + 0.183i)9-s + (−0.445 − 0.895i)12-s + (0.380 − 0.614i)13-s + (0.739 − 0.673i)16-s + (−1.58 − 0.147i)19-s + (−0.850 − 0.526i)21-s + (−0.273 + 0.961i)25-s + (0.273 + 0.961i)27-s + (0.273 − 0.961i)28-s + (0.658 − 0.600i)31-s + (−0.850 + 0.526i)36-s + (−0.942 + 0.469i)37-s + ⋯
L(s)  = 1  + (−0.0922 − 0.995i)3-s + (0.932 − 0.361i)4-s + (0.602 − 0.798i)7-s + (−0.982 + 0.183i)9-s + (−0.445 − 0.895i)12-s + (0.380 − 0.614i)13-s + (0.739 − 0.673i)16-s + (−1.58 − 0.147i)19-s + (−0.850 − 0.526i)21-s + (−0.273 + 0.961i)25-s + (0.273 + 0.961i)27-s + (0.273 − 0.961i)28-s + (0.658 − 0.600i)31-s + (−0.850 + 0.526i)36-s + (−0.942 + 0.469i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2163\)    =    \(3 \cdot 7 \cdot 103\)
Sign: $-0.226 + 0.973i$
Analytic conductor: \(1.07947\)
Root analytic conductor: \(1.03897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2163} (713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2163,\ (\ :0),\ -0.226 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.429910695\)
\(L(\frac12)\) \(\approx\) \(1.429910695\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0922 + 0.995i)T \)
7 \( 1 + (-0.602 + 0.798i)T \)
103 \( 1 + (-0.445 - 0.895i)T \)
good2 \( 1 + (-0.932 + 0.361i)T^{2} \)
5 \( 1 + (0.273 - 0.961i)T^{2} \)
11 \( 1 + (0.932 - 0.361i)T^{2} \)
13 \( 1 + (-0.380 + 0.614i)T + (-0.445 - 0.895i)T^{2} \)
17 \( 1 + (-0.850 - 0.526i)T^{2} \)
19 \( 1 + (1.58 + 0.147i)T + (0.982 + 0.183i)T^{2} \)
23 \( 1 + (-0.932 - 0.361i)T^{2} \)
29 \( 1 + (0.273 - 0.961i)T^{2} \)
31 \( 1 + (-0.658 + 0.600i)T + (0.0922 - 0.995i)T^{2} \)
37 \( 1 + (0.942 - 0.469i)T + (0.602 - 0.798i)T^{2} \)
41 \( 1 + (-0.273 - 0.961i)T^{2} \)
43 \( 1 + (-1.72 - 0.857i)T + (0.602 + 0.798i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.982 - 0.183i)T^{2} \)
59 \( 1 + (0.445 - 0.895i)T^{2} \)
61 \( 1 + (-1.29 - 0.368i)T + (0.850 + 0.526i)T^{2} \)
67 \( 1 + (-0.193 - 0.312i)T + (-0.445 + 0.895i)T^{2} \)
71 \( 1 + (-0.273 - 0.961i)T^{2} \)
73 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
79 \( 1 + (1.02 + 1.35i)T + (-0.273 + 0.961i)T^{2} \)
83 \( 1 + (0.445 + 0.895i)T^{2} \)
89 \( 1 + (-0.739 - 0.673i)T^{2} \)
97 \( 1 + (1.85 - 0.526i)T + (0.850 - 0.526i)T^{2} \)
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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

−8.779029115663503763973045003736, −8.046863047183578655892738070204, −7.43165854092051252227952196081, −6.76064983044013704306202489433, −6.05752250956844330471877356986, −5.31601540163507561095136196548, −4.12409526958558715246801541667, −2.90611230355322660523178697246, −1.96426335426359650664164895420, −1.03371265760591361253316258439, 1.95532398414091706463957340081, 2.69742704798625843068216823379, 3.85139024881034667335615519393, 4.53070077377856734309961352183, 5.63208028040848416212129752103, 6.21584771695515109407970505876, 7.06097592629998695027770755738, 8.358518236073680126959276795656, 8.478446013220935249612414423563, 9.433666032268197955754227622861

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