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Properties

Label 2-432-144.13-c1-0-15
Degree $2$
Conductor $432$
Sign $0.0436 + 0.999i$
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 − i)2-s − 2i·4-s + (0.133 + 0.5i)5-s + (2.13 − 1.23i)7-s + (−2 − 2i)8-s + (0.633 + 0.366i)10-s + (0.5 + 0.133i)11-s + (4.59 − 1.23i)13-s + (0.901 − 3.36i)14-s − 4·16-s − 4·17-s + (−3 − 3i)19-s + (1 − 0.267i)20-s + (0.633 − 0.366i)22-s + (0.401 + 0.232i)23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (0.0599 + 0.223i)5-s + (0.806 − 0.465i)7-s + (−0.707 − 0.707i)8-s + (0.200 + 0.115i)10-s + (0.150 + 0.0403i)11-s + (1.27 − 0.341i)13-s + (0.241 − 0.899i)14-s − 16-s − 0.970·17-s + (−0.688 − 0.688i)19-s + (0.223 − 0.0599i)20-s + (0.135 − 0.0780i)22-s + (0.0838 + 0.0483i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0436 + 0.999i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0436 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53426 - 1.46873i\)
\(L(\frac12)\) \(\approx\) \(1.53426 - 1.46873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
good5 \( 1 + (-0.133 - 0.5i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.13 + 1.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.133i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.59 + 1.23i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (-0.401 - 0.232i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.866 - 3.23i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.598 + 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.73 - 7.73i)T - 37iT^{2} \)
41 \( 1 + (-9.69 - 5.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.69 - 2.33i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.26 - 2.26i)T - 53iT^{2} \)
59 \( 1 + (-1.5 - 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.86 - 14.4i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.23 + 0.330i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.16 - 11.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.85038914651706271754813257857, −10.60532063596027598447138417262, −9.199098661534823205989293438068, −8.374220215361075139299570888491, −6.94361858775481856162498027275, −6.10506541828453617118233986961, −4.84857110397660473702567812082, −4.05309217794302992536761349981, −2.73817215348547152477752612920, −1.28215177371447710200911774828, 2.06062063676510350442717570419, 3.71251742116378939647714884110, 4.64447216272210668371467189189, 5.71325454549113683281076716226, 6.50152729609205577339568083303, 7.65074202273372157566079741528, 8.629821109729049739532567959910, 9.063486702767007750605709710312, 10.86927752403998098214290499492, 11.37385124117238564184406445428

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