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Properties

Label 2-432-27.4-c1-0-8
Degree $2$
Conductor $432$
Sign $0.478 + 0.878i$
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  + (−0.824 − 1.52i)3-s + (−0.265 + 1.50i)5-s + (3.27 − 2.75i)7-s + (−1.64 + 2.51i)9-s + (0.469 + 2.66i)11-s + (3.23 − 1.17i)13-s + (2.51 − 0.836i)15-s + (2.63 − 4.55i)17-s + (−3.51 − 6.09i)19-s + (−6.89 − 2.72i)21-s + (−0.888 − 0.745i)23-s + (2.50 + 0.910i)25-s + (5.17 + 0.429i)27-s + (−0.981 − 0.357i)29-s + (−4.26 − 3.57i)31-s + ⋯
L(s)  = 1  + (−0.475 − 0.879i)3-s + (−0.118 + 0.673i)5-s + (1.23 − 1.03i)7-s + (−0.547 + 0.837i)9-s + (0.141 + 0.802i)11-s + (0.896 − 0.326i)13-s + (0.648 − 0.216i)15-s + (0.638 − 1.10i)17-s + (−0.806 − 1.39i)19-s + (−1.50 − 0.595i)21-s + (−0.185 − 0.155i)23-s + (0.500 + 0.182i)25-s + (0.996 + 0.0826i)27-s + (−0.182 − 0.0663i)29-s + (−0.765 − 0.642i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.478 + 0.878i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.478 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13313 - 0.672932i\)
\(L(\frac12)\) \(\approx\) \(1.13313 - 0.672932i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.824 + 1.52i)T \)
good5 \( 1 + (0.265 - 1.50i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-3.27 + 2.75i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-0.469 - 2.66i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-3.23 + 1.17i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.63 + 4.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.51 + 6.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.888 + 0.745i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.981 + 0.357i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.26 + 3.57i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (0.0292 - 0.0506i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.59 + 3.49i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.26 - 7.18i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-4.71 + 3.95i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 8.72T + 53T^{2} \)
59 \( 1 + (1.90 - 10.8i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.25 - 3.57i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-9.30 + 3.38i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (6.09 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.371 + 0.643i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.26 - 0.460i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-3.87 - 1.41i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.28 - 3.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.745 - 4.22i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.01148707479663947391762572070, −10.57938185673695487453455293156, −9.102430036564264537644989987085, −7.82380295894349734806930535034, −7.36016871221690568008527946536, −6.54326502487683117052415185828, −5.24294045070926225807047559955, −4.22954006162579921799296606502, −2.52328481735338353769945526254, −1.04995375876296575754170714406, 1.56646619708536029852320152488, 3.56991366409868309734131725030, 4.54359156562140779373084942985, 5.63742322460819396491043586593, 6.08853762811353071801799293209, 8.138618643875748965174180525061, 8.547953070663862351598597855336, 9.358945452689761905084884390329, 10.73294427265098267082559258222, 11.09309144781620159443085836349

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