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Properties

Label 2-432-108.95-c1-0-6
Degree $2$
Conductor $432$
Sign $0.435 - 0.900i$
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  + (1.70 − 0.309i)3-s + (−1.45 + 1.73i)5-s + (−1.49 + 4.10i)7-s + (2.80 − 1.05i)9-s + (1.83 − 1.54i)11-s + (−0.219 + 1.24i)13-s + (−1.94 + 3.40i)15-s + (4.23 + 2.44i)17-s + (−5.93 + 3.42i)19-s + (−1.27 + 7.45i)21-s + (−2.43 + 0.885i)23-s + (−0.0202 − 0.114i)25-s + (4.46 − 2.66i)27-s + (9.76 − 1.72i)29-s + (−2.15 − 5.91i)31-s + ⋯
L(s)  = 1  + (0.983 − 0.178i)3-s + (−0.650 + 0.774i)5-s + (−0.564 + 1.55i)7-s + (0.936 − 0.351i)9-s + (0.553 − 0.464i)11-s + (−0.0608 + 0.345i)13-s + (−0.501 + 0.878i)15-s + (1.02 + 0.593i)17-s + (−1.36 + 0.786i)19-s + (−0.278 + 1.62i)21-s + (−0.507 + 0.184i)23-s + (−0.00404 − 0.0229i)25-s + (0.858 − 0.512i)27-s + (1.81 − 0.319i)29-s + (−0.386 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.435 - 0.900i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38668 + 0.869449i\)
\(L(\frac12)\) \(\approx\) \(1.38668 + 0.869449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.309i)T \)
good5 \( 1 + (1.45 - 1.73i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.49 - 4.10i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.83 + 1.54i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.219 - 1.24i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.23 - 2.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.93 - 3.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 - 0.885i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-9.76 + 1.72i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (2.15 + 5.91i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.49 + 4.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.206 - 0.0364i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.863 - 1.02i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-5.30 - 1.92i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 + (6.16 + 5.17i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (3.59 + 1.30i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.97 - 0.525i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.29 + 2.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.41 - 7.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 - 0.245i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.55 + 8.82i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (15.4 - 8.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.8 + 9.98i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.45880247089219715024511216120, −10.24754786856078043849961249290, −9.392361955066434924887821200358, −8.497946898693727129475085248086, −7.88487215813675594158021767254, −6.60779257815908874839630481459, −5.91945132989940502788552117717, −4.06089590416010046335713276451, −3.18970088014545897042482965203, −2.13068844446310127016938419632, 1.01351426432019720052210487981, 3.00179184258271350959936924029, 4.14956469756620712709239849669, 4.65178801451809941107000948692, 6.64274007153779813955319036070, 7.41913870883790614038143948170, 8.266755151011347566119409537843, 9.109162427264404448111020942355, 10.12002724801066682212820952412, 10.64529918211749458642057035306

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