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Properties

Label 2-432-432.131-c1-0-42
Degree $2$
Conductor $432$
Sign $-0.0502 - 0.998i$
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  + (0.905 + 1.08i)2-s + (0.666 + 1.59i)3-s + (−0.361 + 1.96i)4-s + (2.32 − 1.62i)5-s + (−1.13 + 2.17i)6-s + (2.52 − 2.12i)7-s + (−2.46 + 1.38i)8-s + (−2.11 + 2.12i)9-s + (3.86 + 1.05i)10-s + (1.80 + 1.26i)11-s + (−3.38 + 0.732i)12-s + (−0.0257 − 0.0119i)13-s + (4.59 + 0.826i)14-s + (4.14 + 2.62i)15-s + (−3.73 − 1.42i)16-s + (−2.73 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.640 + 0.768i)2-s + (0.384 + 0.923i)3-s + (−0.180 + 0.983i)4-s + (1.03 − 0.726i)5-s + (−0.463 + 0.886i)6-s + (0.955 − 0.802i)7-s + (−0.871 + 0.490i)8-s + (−0.704 + 0.709i)9-s + (1.22 + 0.332i)10-s + (0.544 + 0.381i)11-s + (−0.977 + 0.211i)12-s + (−0.00713 − 0.00332i)13-s + (1.22 + 0.220i)14-s + (1.06 + 0.678i)15-s + (−0.934 − 0.355i)16-s + (−0.664 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0502 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0502 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.0502 - 0.998i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.0502 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73748 + 1.82711i\)
\(L(\frac12)\) \(\approx\) \(1.73748 + 1.82711i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.905 - 1.08i)T \)
3 \( 1 + (-0.666 - 1.59i)T \)
good5 \( 1 + (-2.32 + 1.62i)T + (1.71 - 4.69i)T^{2} \)
7 \( 1 + (-2.52 + 2.12i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-1.80 - 1.26i)T + (3.76 + 10.3i)T^{2} \)
13 \( 1 + (0.0257 + 0.0119i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (2.73 + 1.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.23 + 1.13i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.41 + 4.07i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (8.52 - 3.97i)T + (18.6 - 22.2i)T^{2} \)
31 \( 1 + (-0.504 + 0.600i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (1.40 + 5.25i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (9.20 - 3.35i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.740 + 1.05i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-1.24 + 1.04i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-8.44 - 8.44i)T + 53iT^{2} \)
59 \( 1 + (-4.76 - 6.80i)T + (-20.1 + 55.4i)T^{2} \)
61 \( 1 + (3.61 + 0.316i)T + (60.0 + 10.5i)T^{2} \)
67 \( 1 + (-4.78 + 10.2i)T + (-43.0 - 51.3i)T^{2} \)
71 \( 1 + (3.23 + 1.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-11.3 + 6.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.44 + 6.72i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-7.13 - 15.2i)T + (-53.3 + 63.5i)T^{2} \)
89 \( 1 + (-4.06 - 7.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.04 - 5.95i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.27695342207984555920904388270, −10.49885243639549689200028794233, −9.163234908951255335009512882336, −8.871658122061461537771632810986, −7.71271154928680846840862987963, −6.61373490245089363807699480085, −5.32843652874980007963159338890, −4.72300508841200121094924537887, −3.88441140407188858715436912851, −2.15460837965909518030163387803, 1.73958150462243979918075571773, 2.31216345390937244943966173870, 3.64107512559940596334216892175, 5.30212097430465310038970491089, 6.09515207114886668731421033109, 6.87259201701866101102026744495, 8.406456313768991132450713031888, 9.125755107502991593519666333148, 10.16585574777274722261163152665, 11.30194305496006622863115018869

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