Properties

Label 2-432-108.11-c1-0-5
Degree $2$
Conductor $432$
Sign $0.948 + 0.318i$
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  + (−1.67 − 0.450i)3-s + (−0.436 + 1.19i)5-s + (−3.53 − 0.622i)7-s + (2.59 + 1.50i)9-s + (4.59 − 1.67i)11-s + (1.75 − 1.47i)13-s + (1.27 − 1.80i)15-s + (0.393 − 0.227i)17-s + (5.43 + 3.13i)19-s + (5.62 + 2.63i)21-s + (−0.629 − 3.57i)23-s + (2.58 + 2.16i)25-s + (−3.66 − 3.68i)27-s + (6.09 − 7.26i)29-s + (0.352 − 0.0621i)31-s + ⋯
L(s)  = 1  + (−0.965 − 0.260i)3-s + (−0.195 + 0.536i)5-s + (−1.33 − 0.235i)7-s + (0.864 + 0.502i)9-s + (1.38 − 0.504i)11-s + (0.485 − 0.407i)13-s + (0.327 − 0.467i)15-s + (0.0954 − 0.0551i)17-s + (1.24 + 0.719i)19-s + (1.22 + 0.574i)21-s + (−0.131 − 0.744i)23-s + (0.516 + 0.433i)25-s + (−0.704 − 0.709i)27-s + (1.13 − 1.34i)29-s + (0.0633 − 0.0111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.948 + 0.318i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.948 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926601 - 0.151286i\)
\(L(\frac12)\) \(\approx\) \(0.926601 - 0.151286i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 0.450i)T \)
good5 \( 1 + (0.436 - 1.19i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (3.53 + 0.622i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-4.59 + 1.67i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.75 + 1.47i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.393 + 0.227i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.43 - 3.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.629 + 3.57i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.09 + 7.26i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.352 + 0.0621i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.46 - 4.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.66 + 5.56i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.37 - 6.52i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.475 + 2.69i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.30iT - 53T^{2} \)
59 \( 1 + (-9.78 - 3.56i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.68 + 15.2i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.26 - 1.50i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.92 - 5.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.44 - 5.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.89 - 7.02i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.97 + 8.37i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-0.480 - 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.40 - 1.96i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.24084754379778853828803267528, −10.18791594075322648241556401626, −9.640220994646613808050622584491, −8.279368654059786627119662191487, −7.01309581126718229406150300420, −6.46246756611607821882988321505, −5.69593716600455819481629146559, −4.11629361669669946934994852042, −3.12606665801225466184446225407, −0.929539520172569096110741704084, 1.09687424730759802170543649448, 3.37099573448029210315853799612, 4.39791447873559504464013317528, 5.50553985483629939694060020553, 6.56193464157677750263542484021, 7.07448385962457584609867518567, 8.838584853726952182750810578144, 9.439480241112073769613595213716, 10.19490871644768359627335817252, 11.41221638959671778877856688151

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