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Properties

Label 2-432-16.13-c1-0-12
Degree $2$
Conductor $432$
Sign $0.738 - 0.673i$
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  + (−0.680 + 1.23i)2-s + (−1.07 − 1.68i)4-s + (−2.24 + 2.24i)5-s − 3.93i·7-s + (2.82 − 0.180i)8-s + (−1.25 − 4.31i)10-s + (3.47 − 3.47i)11-s + (4.14 + 4.14i)13-s + (4.87 + 2.67i)14-s + (−1.69 + 3.62i)16-s + 0.326·17-s + (−0.108 − 0.108i)19-s + (6.20 + 1.38i)20-s + (1.94 + 6.67i)22-s + 4.30i·23-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (−0.536 − 0.843i)4-s + (−1.00 + 1.00i)5-s − 1.48i·7-s + (0.997 − 0.0637i)8-s + (−0.396 − 1.36i)10-s + (1.04 − 1.04i)11-s + (1.14 + 1.14i)13-s + (1.30 + 0.716i)14-s + (−0.424 + 0.905i)16-s + 0.0792·17-s + (−0.0249 − 0.0249i)19-s + (1.38 + 0.309i)20-s + (0.413 + 1.42i)22-s + 0.897i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.738 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882702 + 0.342120i\)
\(L(\frac12)\) \(\approx\) \(0.882702 + 0.342120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.680 - 1.23i)T \)
3 \( 1 \)
good5 \( 1 + (2.24 - 2.24i)T - 5iT^{2} \)
7 \( 1 + 3.93iT - 7T^{2} \)
11 \( 1 + (-3.47 + 3.47i)T - 11iT^{2} \)
13 \( 1 + (-4.14 - 4.14i)T + 13iT^{2} \)
17 \( 1 - 0.326T + 17T^{2} \)
19 \( 1 + (0.108 + 0.108i)T + 19iT^{2} \)
23 \( 1 - 4.30iT - 23T^{2} \)
29 \( 1 + (-2.75 - 2.75i)T + 29iT^{2} \)
31 \( 1 - 4.45T + 31T^{2} \)
37 \( 1 + (-4.76 + 4.76i)T - 37iT^{2} \)
41 \( 1 + 9.19iT - 41T^{2} \)
43 \( 1 + (-5.45 + 5.45i)T - 43iT^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 + (2.59 - 2.59i)T - 53iT^{2} \)
59 \( 1 + (-0.326 + 0.326i)T - 59iT^{2} \)
61 \( 1 + (-8.26 - 8.26i)T + 61iT^{2} \)
67 \( 1 + (-4.31 - 4.31i)T + 67iT^{2} \)
71 \( 1 + 6.88iT - 71T^{2} \)
73 \( 1 + 3.15iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + (1.07 + 1.07i)T + 83iT^{2} \)
89 \( 1 - 5.77iT - 89T^{2} \)
97 \( 1 - 2.82T + 97T^{2} \)
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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

−11.01564305294615837364952223176, −10.52321196837946392171260325667, −9.250179119545094179262446516382, −8.396879061540354477843235786672, −7.33818567745708609520226422104, −6.87848787398551140445231078126, −5.99102648735254090695680891484, −4.12259795507826604355771673699, −3.72560521183584203675202174878, −1.00605039294739502797830827431, 1.11890227276894231624217471061, 2.73121388499844393795819696295, 4.05716574470563265079894259133, 4.90677523344854291859214240936, 6.32681689442847252995570315074, 7.985975541334035136573852045709, 8.395673818886876586755600713512, 9.176067788542175236324827634193, 10.03444085738843301871311498250, 11.38108804850764691140022891982

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