Properties

Label 2-432-16.13-c1-0-1
Degree $2$
Conductor $432$
Sign $-0.615 + 0.788i$
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  + (0.451 + 1.34i)2-s + (−1.59 + 1.21i)4-s + (−2.86 + 2.86i)5-s − 1.20i·7-s + (−2.34 − 1.58i)8-s + (−5.13 − 2.54i)10-s + (0.703 − 0.703i)11-s + (−2.30 − 2.30i)13-s + (1.61 − 0.543i)14-s + (1.06 − 3.85i)16-s − 5.47·17-s + (3.25 + 3.25i)19-s + (1.08 − 8.02i)20-s + (1.26 + 0.624i)22-s + 2.74i·23-s + ⋯
L(s)  = 1  + (0.319 + 0.947i)2-s + (−0.795 + 0.605i)4-s + (−1.28 + 1.28i)5-s − 0.454i·7-s + (−0.828 − 0.560i)8-s + (−1.62 − 0.804i)10-s + (0.212 − 0.212i)11-s + (−0.639 − 0.639i)13-s + (0.430 − 0.145i)14-s + (0.266 − 0.963i)16-s − 1.32·17-s + (0.746 + 0.746i)19-s + (0.243 − 1.79i)20-s + (0.268 + 0.133i)22-s + 0.571i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194512 - 0.398411i\)
\(L(\frac12)\) \(\approx\) \(0.194512 - 0.398411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.451 - 1.34i)T \)
3 \( 1 \)
good5 \( 1 + (2.86 - 2.86i)T - 5iT^{2} \)
7 \( 1 + 1.20iT - 7T^{2} \)
11 \( 1 + (-0.703 + 0.703i)T - 11iT^{2} \)
13 \( 1 + (2.30 + 2.30i)T + 13iT^{2} \)
17 \( 1 + 5.47T + 17T^{2} \)
19 \( 1 + (-3.25 - 3.25i)T + 19iT^{2} \)
23 \( 1 - 2.74iT - 23T^{2} \)
29 \( 1 + (-4.25 - 4.25i)T + 29iT^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + (4.02 - 4.02i)T - 37iT^{2} \)
41 \( 1 + 0.790iT - 41T^{2} \)
43 \( 1 + (4.88 - 4.88i)T - 43iT^{2} \)
47 \( 1 + 9.29T + 47T^{2} \)
53 \( 1 + (4.54 - 4.54i)T - 53iT^{2} \)
59 \( 1 + (2.89 - 2.89i)T - 59iT^{2} \)
61 \( 1 + (-1.64 - 1.64i)T + 61iT^{2} \)
67 \( 1 + (-7.66 - 7.66i)T + 67iT^{2} \)
71 \( 1 - 1.62iT - 71T^{2} \)
73 \( 1 + 1.69iT - 73T^{2} \)
79 \( 1 - 8.99T + 79T^{2} \)
83 \( 1 + (-3.57 - 3.57i)T + 83iT^{2} \)
89 \( 1 + 17.3iT - 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−11.73196270494931019472171615217, −10.92555649301554755074891228371, −9.957825131102803205745560324736, −8.678476406438677289363437422601, −7.70546781149698299203484680352, −7.18863272378703649101555746683, −6.39339053735157447867319984848, −5.01781221596686413734038086058, −3.84163418846816712911986852971, −3.12172118596406882545948993589, 0.24879829659274766473104385435, 2.03649417339012774958715769349, 3.64871122860735790229219493894, 4.60566865075761292652894114926, 5.18325567844696555533368660666, 6.83968377504126205987126254232, 8.142164411680008744416642262992, 8.996130945993670526512716013336, 9.458121130775601514926693490608, 10.95061625801134461511825248831

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