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Properties

Label 2-432-12.11-c1-0-1
Degree $2$
Conductor $432$
Sign $-i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19i·7-s − 5·13-s + 5.19i·19-s + 5·25-s + 10.3i·31-s + 11·37-s − 10.3i·43-s − 20·49-s − 61-s − 15.5i·67-s + 7·73-s + 5.19i·79-s − 25.9i·91-s + 19·97-s − 15.5i·103-s + ⋯
L(s)  = 1  + 1.96i·7-s − 1.38·13-s + 1.19i·19-s + 25-s + 1.86i·31-s + 1.80·37-s − 1.58i·43-s − 2.85·49-s − 0.128·61-s − 1.90i·67-s + 0.819·73-s + 0.584i·79-s − 2.72i·91-s + 1.92·97-s − 1.53i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822021 + 0.822021i\)
\(L(\frac12)\) \(\approx\) \(0.822021 + 0.822021i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 5.19iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 15.5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−11.58386422465684181661120154846, −10.40092110870909484542111328135, −9.460037139390836583475419601217, −8.747607751506310630572770261150, −7.83973002046380619543041672981, −6.60817338763161248111332032538, −5.58479572145862521463143700297, −4.84219169406570477186014901126, −3.10183920413156191538396756360, −2.08031965102448241621398582936, 0.74084876375293433558861635527, 2.70999451656945664170355507599, 4.17075201709690677704368048402, 4.83791216107130437830044645588, 6.44016159785880704463838000079, 7.32032889076460286744362461331, 7.86832013851592462962153361050, 9.377707646825199205050485379873, 10.01499262651823060439340749546, 10.92514051473156948034913745098

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