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Properties

Label 2-2163-2163.2099-c0-0-1
Degree $2$
Conductor $2163$
Sign $0.910 + 0.413i$
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  + (0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.273 + 0.961i)7-s + (0.932 − 0.361i)9-s + (0.602 − 0.798i)12-s + (−1.20 + 0.600i)13-s + (0.0922 − 0.995i)16-s + (0.353 − 1.89i)19-s + (0.445 + 0.895i)21-s + (−0.850 − 0.526i)25-s + (0.850 − 0.526i)27-s + (0.850 + 0.526i)28-s + (−0.111 + 1.20i)31-s + (0.445 − 0.895i)36-s + (1.42 + 1.07i)37-s + ⋯
L(s)  = 1  + (0.982 − 0.183i)3-s + (0.739 − 0.673i)4-s + (0.273 + 0.961i)7-s + (0.932 − 0.361i)9-s + (0.602 − 0.798i)12-s + (−1.20 + 0.600i)13-s + (0.0922 − 0.995i)16-s + (0.353 − 1.89i)19-s + (0.445 + 0.895i)21-s + (−0.850 − 0.526i)25-s + (0.850 − 0.526i)27-s + (0.850 + 0.526i)28-s + (−0.111 + 1.20i)31-s + (0.445 − 0.895i)36-s + (1.42 + 1.07i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2163\)    =    \(3 \cdot 7 \cdot 103\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(1.07947\)
Root analytic conductor: \(1.03897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2163} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2163,\ (\ :0),\ 0.910 + 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952358334\)
\(L(\frac12)\) \(\approx\) \(1.952358334\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (-0.273 - 0.961i)T \)
103 \( 1 + (0.602 - 0.798i)T \)
good2 \( 1 + (-0.739 + 0.673i)T^{2} \)
5 \( 1 + (0.850 + 0.526i)T^{2} \)
11 \( 1 + (0.739 - 0.673i)T^{2} \)
13 \( 1 + (1.20 - 0.600i)T + (0.602 - 0.798i)T^{2} \)
17 \( 1 + (0.445 + 0.895i)T^{2} \)
19 \( 1 + (-0.353 + 1.89i)T + (-0.932 - 0.361i)T^{2} \)
23 \( 1 + (-0.739 - 0.673i)T^{2} \)
29 \( 1 + (0.850 + 0.526i)T^{2} \)
31 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
37 \( 1 + (-1.42 - 1.07i)T + (0.273 + 0.961i)T^{2} \)
41 \( 1 + (-0.850 + 0.526i)T^{2} \)
43 \( 1 + (0.840 - 0.634i)T + (0.273 - 0.961i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.932 + 0.361i)T^{2} \)
59 \( 1 + (-0.602 - 0.798i)T^{2} \)
61 \( 1 + (1.04 - 1.69i)T + (-0.445 - 0.895i)T^{2} \)
67 \( 1 + (-0.646 - 0.322i)T + (0.602 + 0.798i)T^{2} \)
71 \( 1 + (-0.850 + 0.526i)T^{2} \)
73 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
79 \( 1 + (-0.243 + 0.857i)T + (-0.850 - 0.526i)T^{2} \)
83 \( 1 + (-0.602 + 0.798i)T^{2} \)
89 \( 1 + (-0.0922 - 0.995i)T^{2} \)
97 \( 1 + (0.554 + 0.895i)T + (-0.445 + 0.895i)T^{2} \)
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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

−9.336396418596987479142004078016, −8.485846394690486670822097239031, −7.60471544557926157226370549705, −6.93208897415110967833359935730, −6.26531661334306975302803835015, −5.12049135863621101330987851376, −4.51399097884978383564197008745, −2.87614571572879285366463243536, −2.52011004995574372341674730769, −1.49398126880339318476603528243, 1.67953219970409074252965996979, 2.55995047179890474791292765405, 3.63315453653994545974446000117, 4.03861042750205928542928406343, 5.25609215835048243403760688183, 6.38031814254231224761610270485, 7.43294426154672587542441956955, 7.75100720208732656734229354651, 8.149068404588320275163895777972, 9.450709724361676653345864340502

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