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Properties

Label 2-432-27.16-c1-0-7
Degree $2$
Conductor $432$
Sign $0.988 + 0.151i$
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  + (1.36 − 1.07i)3-s + (0.696 + 0.253i)5-s + (−0.717 + 4.07i)7-s + (0.703 − 2.91i)9-s + (4.27 − 1.55i)11-s + (0.662 − 0.556i)13-s + (1.21 − 0.401i)15-s + (2.17 + 3.77i)17-s + (0.777 − 1.34i)19-s + (3.38 + 6.30i)21-s + (0.608 + 3.45i)23-s + (−3.40 − 2.86i)25-s + (−2.16 − 4.72i)27-s + (−2.50 − 2.10i)29-s + (−1.85 − 10.5i)31-s + ⋯
L(s)  = 1  + (0.785 − 0.618i)3-s + (0.311 + 0.113i)5-s + (−0.271 + 1.53i)7-s + (0.234 − 0.972i)9-s + (1.28 − 0.468i)11-s + (0.183 − 0.154i)13-s + (0.314 − 0.103i)15-s + (0.528 + 0.915i)17-s + (0.178 − 0.309i)19-s + (0.738 + 1.37i)21-s + (0.126 + 0.719i)23-s + (−0.681 − 0.572i)25-s + (−0.417 − 0.908i)27-s + (−0.464 − 0.390i)29-s + (−0.333 − 1.89i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92778 - 0.146410i\)
\(L(\frac12)\) \(\approx\) \(1.92778 - 0.146410i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.07i)T \)
good5 \( 1 + (-0.696 - 0.253i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (0.717 - 4.07i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-4.27 + 1.55i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.662 + 0.556i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.17 - 3.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.777 + 1.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.608 - 3.45i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (2.50 + 2.10i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.85 + 10.5i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-0.880 - 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.97 - 1.65i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.58 - 0.941i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.68 - 9.54i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
59 \( 1 + (-1.34 - 0.489i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.751 - 4.26i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (10.0 - 8.42i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.54 + 4.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.286 + 0.496i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.17 + 4.34i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.06 + 5.92i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.19 + 10.7i)T + (-44.5 - 77.0i)T^{2} \)
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

−11.50868799445470326290547267794, −9.856522098143750604388314284409, −9.212497833367299652403678552410, −8.522543432417433061583605367497, −7.57665233906540097625626286685, −6.16525867439362286967871019912, −5.91181841835686968124801114148, −3.94066619675452973401831284416, −2.83351768009256821410317735415, −1.65871973172107526798145154584, 1.52839090323605257612022635418, 3.36292797177033876798745477579, 4.08250087691380123465423611019, 5.16783140352999637855177745560, 6.81102385019865514608570450142, 7.37851431750060323460989533661, 8.630380558699629451201088463236, 9.492512418548855941417544448223, 10.10793178905834127949991864778, 10.90795724765172414391270603356

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