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Properties

Label 2-432-432.131-c1-0-22
Degree $2$
Conductor $432$
Sign $-0.217 - 0.976i$
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  + (1.08 + 0.910i)2-s + (1.27 − 1.16i)3-s + (0.340 + 1.97i)4-s + (−2.98 + 2.09i)5-s + (2.44 − 0.0968i)6-s + (−2.26 + 1.89i)7-s + (−1.42 + 2.44i)8-s + (0.275 − 2.98i)9-s + (−5.13 − 0.458i)10-s + (4.73 + 3.31i)11-s + (2.73 + 2.12i)12-s + (2.43 + 1.13i)13-s + (−4.17 − 0.00689i)14-s + (−1.38 + 6.16i)15-s + (−3.76 + 1.34i)16-s + (2.13 + 1.23i)17-s + ⋯
L(s)  = 1  + (0.764 + 0.644i)2-s + (0.738 − 0.673i)3-s + (0.170 + 0.985i)4-s + (−1.33 + 0.935i)5-s + (0.999 − 0.0395i)6-s + (−0.855 + 0.717i)7-s + (−0.504 + 0.863i)8-s + (0.0919 − 0.995i)9-s + (−1.62 − 0.144i)10-s + (1.42 + 0.998i)11-s + (0.789 + 0.613i)12-s + (0.674 + 0.314i)13-s + (−1.11 − 0.00184i)14-s + (−0.356 + 1.59i)15-s + (−0.941 + 0.335i)16-s + (0.517 + 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24397 + 1.55100i\)
\(L(\frac12)\) \(\approx\) \(1.24397 + 1.55100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.910i)T \)
3 \( 1 + (-1.27 + 1.16i)T \)
good5 \( 1 + (2.98 - 2.09i)T + (1.71 - 4.69i)T^{2} \)
7 \( 1 + (2.26 - 1.89i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-4.73 - 3.31i)T + (3.76 + 10.3i)T^{2} \)
13 \( 1 + (-2.43 - 1.13i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (-2.13 - 1.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.66 + 1.25i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.67 + 4.38i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.42 + 1.59i)T + (18.6 - 22.2i)T^{2} \)
31 \( 1 + (3.68 - 4.39i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (0.773 + 2.88i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.53 + 0.923i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-4.10 + 5.86i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (-2.31 + 1.94i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-4.36 - 4.36i)T + 53iT^{2} \)
59 \( 1 + (-4.25 - 6.07i)T + (-20.1 + 55.4i)T^{2} \)
61 \( 1 + (-3.96 - 0.346i)T + (60.0 + 10.5i)T^{2} \)
67 \( 1 + (-5.70 + 12.2i)T + (-43.0 - 51.3i)T^{2} \)
71 \( 1 + (8.69 + 5.01i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.4 - 6.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.85 + 5.08i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.07 - 10.8i)T + (-53.3 + 63.5i)T^{2} \)
89 \( 1 + (0.108 + 0.187i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.04 - 17.2i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.99963534248353864499628221430, −10.75255179125771516017231435907, −9.096668678579056173990750539212, −8.568809351516507256377874099960, −7.38223472962101200183670952751, −6.78989819408637567944065321580, −6.22699746896976838667437900579, −4.19860403344158236870910741154, −3.57352750189642156260779889814, −2.49174761239390477526038834937, 0.994399411659667745138711653910, 3.33707394577851972225308627734, 3.76288367622832313936820455260, 4.52731333795532481239365418629, 5.89674371886024554595510195634, 7.19778164306408206528619573007, 8.462446887626882961733695088076, 9.111302353491912226013228890320, 10.06472100390585038457639208656, 11.13213370812781449659741739551

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