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Properties

Label 2-432-16.5-c1-0-3
Degree $2$
Conductor $432$
Sign $0.923 - 0.382i$
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  − 1.41i·2-s − 2.00·4-s + (0.707 + 0.707i)5-s + 3i·7-s + 2.82i·8-s + (1.00 − 1.00i)10-s + (0.707 + 0.707i)11-s + (−5 + 5i)13-s + 4.24·14-s + 4.00·16-s + 4.24·17-s + (−2 + 2i)19-s + (−1.41 − 1.41i)20-s + (1.00 − 1.00i)22-s + 5.65i·23-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (0.316 + 0.316i)5-s + 1.13i·7-s + 1.00i·8-s + (0.316 − 0.316i)10-s + (0.213 + 0.213i)11-s + (−1.38 + 1.38i)13-s + 1.13·14-s + 1.00·16-s + 1.02·17-s + (−0.458 + 0.458i)19-s + (−0.316 − 0.316i)20-s + (0.213 − 0.213i)22-s + 1.17i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10661 + 0.220120i\)
\(L(\frac12)\) \(\approx\) \(1.10661 + 0.220120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T + 5iT^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + 11iT^{2} \)
13 \( 1 + (5 - 5i)T - 13iT^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + (2 - 2i)T - 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (-5.65 + 5.65i)T - 29iT^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + (-4 - 4i)T + 37iT^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + (-1 - i)T + 43iT^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + (7.77 + 7.77i)T + 53iT^{2} \)
59 \( 1 + (-7.07 - 7.07i)T + 59iT^{2} \)
61 \( 1 + (2 - 2i)T - 61iT^{2} \)
67 \( 1 + (5 - 5i)T - 67iT^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 3iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + (4.94 - 4.94i)T - 83iT^{2} \)
89 \( 1 + 15.5iT - 89T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−11.52195866086626135942146821554, −10.04901448640108471999529155933, −9.724462644707531635849239512415, −8.788646881881624075592312351047, −7.73842040573584713080577347079, −6.34585791945191617557791159134, −5.28219412154020868924249274055, −4.23884246784672676153777627001, −2.78527710980162340662330233998, −1.89194285751952004614765969622, 0.74746319509705047448031245802, 3.20839331520026772562491140943, 4.58124967098889192171986960913, 5.32756047011352069384440388161, 6.50356323943057660843832123400, 7.44723686624247340613608639382, 8.081627367623722765933909078524, 9.232602896862476712465972123932, 10.09307085912228338132741858392, 10.73504110353428198773664187588

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