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Properties

Label 2-432-108.23-c1-0-1
Degree $2$
Conductor $432$
Sign $0.668 - 0.743i$
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  + (−0.991 − 1.42i)3-s + (−3.85 − 0.678i)5-s + (1.90 + 2.26i)7-s + (−1.03 + 2.81i)9-s + (0.560 + 3.17i)11-s + (1.43 − 0.523i)13-s + (2.85 + 6.14i)15-s + (−2.60 − 1.50i)17-s + (4.90 − 2.83i)19-s + (1.33 − 4.94i)21-s + (6.11 + 5.13i)23-s + (9.66 + 3.51i)25-s + (5.02 − 1.32i)27-s + (−1.57 + 4.33i)29-s + (−6.11 + 7.28i)31-s + ⋯
L(s)  = 1  + (−0.572 − 0.820i)3-s + (−1.72 − 0.303i)5-s + (0.719 + 0.856i)7-s + (−0.344 + 0.938i)9-s + (0.168 + 0.957i)11-s + (0.399 − 0.145i)13-s + (0.736 + 1.58i)15-s + (−0.630 − 0.364i)17-s + (1.12 − 0.649i)19-s + (0.291 − 1.08i)21-s + (1.27 + 1.07i)23-s + (1.93 + 0.703i)25-s + (0.967 − 0.254i)27-s + (−0.292 + 0.804i)29-s + (−1.09 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.669527 + 0.298548i\)
\(L(\frac12)\) \(\approx\) \(0.669527 + 0.298548i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.991 + 1.42i)T \)
good5 \( 1 + (3.85 + 0.678i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.90 - 2.26i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.560 - 3.17i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-1.43 + 0.523i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.60 + 1.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.90 + 2.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.11 - 5.13i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.57 - 4.33i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (6.11 - 7.28i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.24 - 3.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.210 - 0.577i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (3.35 - 0.591i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.982 - 0.824i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 5.01iT - 53T^{2} \)
59 \( 1 + (0.0386 - 0.219i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (10.2 - 8.60i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.16 - 3.20i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.12 + 7.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.87 - 6.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.28 - 3.52i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-11.3 - 4.14i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.527 + 0.304i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.21 + 18.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.40124359658238638821290089520, −10.94333625766031271938343007589, −9.141413633671492986004580007139, −8.427146075789013984045493207316, −7.37775177239242471522413128454, −7.03551754475354558848689287011, −5.29515937742347036067773195372, −4.75279289600233084744966270922, −3.16909140238029773343780892208, −1.39607047332078762882196209080, 0.56797423765914602838733164321, 3.46604532226917280568352046435, 4.01015389915333690790668720523, 4.98678590260397965130691365467, 6.34638167188059848727138764483, 7.43918311235464198169693706613, 8.225380954293044615813212193648, 9.175939456230494024612539997569, 10.57582457816196999499569442874, 11.15043823741604309063912668289

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