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Properties

Label 2-432-48.11-c1-0-8
Degree $2$
Conductor $432$
Sign $0.981 - 0.190i$
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  + (0.0710 − 1.41i)2-s + (−1.98 − 0.200i)4-s + (−1.84 + 1.84i)5-s − 0.0355·7-s + (−0.424 + 2.79i)8-s + (2.47 + 2.73i)10-s + (1.59 + 1.59i)11-s + (2.90 − 2.90i)13-s + (−0.00252 + 0.0502i)14-s + (3.91 + 0.798i)16-s + 5.38i·17-s + (5.17 + 5.17i)19-s + (4.04 − 3.30i)20-s + (2.36 − 2.13i)22-s + 3.09i·23-s + ⋯
L(s)  = 1  + (0.0502 − 0.998i)2-s + (−0.994 − 0.100i)4-s + (−0.825 + 0.825i)5-s − 0.0134·7-s + (−0.150 + 0.988i)8-s + (0.783 + 0.865i)10-s + (0.480 + 0.480i)11-s + (0.804 − 0.804i)13-s + (−0.000675 + 0.0134i)14-s + (0.979 + 0.199i)16-s + 1.30i·17-s + (1.18 + 1.18i)19-s + (0.904 − 0.738i)20-s + (0.503 − 0.455i)22-s + 0.644i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.981 - 0.190i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.981 - 0.190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03484 + 0.0995056i\)
\(L(\frac12)\) \(\approx\) \(1.03484 + 0.0995056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0710 + 1.41i)T \)
3 \( 1 \)
good5 \( 1 + (1.84 - 1.84i)T - 5iT^{2} \)
7 \( 1 + 0.0355T + 7T^{2} \)
11 \( 1 + (-1.59 - 1.59i)T + 11iT^{2} \)
13 \( 1 + (-2.90 + 2.90i)T - 13iT^{2} \)
17 \( 1 - 5.38iT - 17T^{2} \)
19 \( 1 + (-5.17 - 5.17i)T + 19iT^{2} \)
23 \( 1 - 3.09iT - 23T^{2} \)
29 \( 1 + (0.221 + 0.221i)T + 29iT^{2} \)
31 \( 1 + 0.437iT - 31T^{2} \)
37 \( 1 + (-1.48 - 1.48i)T + 37iT^{2} \)
41 \( 1 + 7.36T + 41T^{2} \)
43 \( 1 + (7.32 - 7.32i)T - 43iT^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 + (-8.73 + 8.73i)T - 53iT^{2} \)
59 \( 1 + (-7.78 - 7.78i)T + 59iT^{2} \)
61 \( 1 + (-9.41 + 9.41i)T - 61iT^{2} \)
67 \( 1 + (1.21 + 1.21i)T + 67iT^{2} \)
71 \( 1 + 5.36iT - 71T^{2} \)
73 \( 1 + 8.99iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−11.35141541973440408920137271661, −10.32300644468637795349378872106, −9.775484102328787043523339159188, −8.384282361277075372699774173970, −7.80796492734826000489553515932, −6.43938448969630310616116705833, −5.24498683842738371095091802473, −3.72852398807818552899843401726, −3.37391098518399284054685393357, −1.56003634521938118113549833733, 0.73875277588014807774558186260, 3.43212547312427087798180507818, 4.51305213825285163121825445028, 5.28847080276411850948163325988, 6.61932674417113004563379662627, 7.32824973071591247584811975619, 8.538944535666396390041100346412, 8.879223846620002823578573881732, 9.884355246020086858282927852053, 11.49351827018103266525054897532

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