Properties

Label 2-432-27.14-c2-0-20
Degree $2$
Conductor $432$
Sign $0.0555 + 0.998i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.49 − 2.59i)3-s + (−5.00 + 5.96i)5-s + (3.39 + 1.23i)7-s + (−4.51 + 7.78i)9-s + (−2.59 − 3.09i)11-s + (2.31 − 13.1i)13-s + (22.9 + 4.07i)15-s + (20.7 + 11.9i)17-s + (−13.5 − 23.5i)19-s + (−1.87 − 10.6i)21-s + (−3.97 − 10.9i)23-s + (−6.17 − 35.0i)25-s + (26.9 + 0.0701i)27-s + (22.8 − 4.03i)29-s + (3.81 − 1.38i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−1.00 + 1.19i)5-s + (0.484 + 0.176i)7-s + (−0.501 + 0.865i)9-s + (−0.236 − 0.281i)11-s + (0.177 − 1.00i)13-s + (1.53 + 0.271i)15-s + (1.22 + 0.705i)17-s + (−0.715 − 1.23i)19-s + (−0.0891 − 0.508i)21-s + (−0.172 − 0.474i)23-s + (−0.247 − 1.40i)25-s + (0.999 + 0.00259i)27-s + (0.788 − 0.139i)29-s + (0.122 − 0.0447i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0555 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0555 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0555 + 0.998i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ 0.0555 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.670338 - 0.634079i\)
\(L(\frac12)\) \(\approx\) \(0.670338 - 0.634079i\)
\(L(2)\) 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 \( 1 + (1.49 + 2.59i)T \)
good5 \( 1 + (5.00 - 5.96i)T + (-4.34 - 24.6i)T^{2} \)
7 \( 1 + (-3.39 - 1.23i)T + (37.5 + 31.4i)T^{2} \)
11 \( 1 + (2.59 + 3.09i)T + (-21.0 + 119. i)T^{2} \)
13 \( 1 + (-2.31 + 13.1i)T + (-158. - 57.8i)T^{2} \)
17 \( 1 + (-20.7 - 11.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.5 + 23.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (3.97 + 10.9i)T + (-405. + 340. i)T^{2} \)
29 \( 1 + (-22.8 + 4.03i)T + (790. - 287. i)T^{2} \)
31 \( 1 + (-3.81 + 1.38i)T + (736. - 617. i)T^{2} \)
37 \( 1 + (-35.3 + 61.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (43.0 + 7.59i)T + (1.57e3 + 574. i)T^{2} \)
43 \( 1 + (35.3 - 29.6i)T + (321. - 1.82e3i)T^{2} \)
47 \( 1 + (-28.3 + 77.9i)T + (-1.69e3 - 1.41e3i)T^{2} \)
53 \( 1 + 28.9iT - 2.80e3T^{2} \)
59 \( 1 + (33.8 - 40.3i)T + (-604. - 3.42e3i)T^{2} \)
61 \( 1 + (-4.08 - 1.48i)T + (2.85e3 + 2.39e3i)T^{2} \)
67 \( 1 + (-22.7 + 129. i)T + (-4.21e3 - 1.53e3i)T^{2} \)
71 \( 1 + (-60.4 - 34.8i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-65.4 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-4.20 - 23.8i)T + (-5.86e3 + 2.13e3i)T^{2} \)
83 \( 1 + (-41.0 + 7.22i)T + (6.47e3 - 2.35e3i)T^{2} \)
89 \( 1 + (-84.1 + 48.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-37.2 + 31.2i)T + (1.63e3 - 9.26e3i)T^{2} \)
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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

−10.87842300995207360391767971818, −10.28131456359638273601593860293, −8.391525368856408404213223787220, −7.934543656384367796296721147930, −7.02768691485436043348978450082, −6.17277066468877969457454392756, −5.06689884376248526279258670752, −3.52994187661936988483091872895, −2.41231593816042496176734857503, −0.49020872633498211635085148527, 1.18703732149706383805083461375, 3.51526165533636802271538533588, 4.50164546492781094755800908763, 5.02520931331040491406603792618, 6.29365438501800775570584088214, 7.75241263644745155050726097508, 8.415375764594672847926584426400, 9.415583537800859537586689841190, 10.21270885053927842484123433467, 11.33073472781501208898570799006

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