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Properties

Label 2-432-48.35-c1-0-24
Degree $2$
Conductor $432$
Sign $0.579 + 0.814i$
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  + (1.12 − 0.850i)2-s + (0.552 − 1.92i)4-s + (1.26 + 1.26i)5-s + 1.47·7-s + (−1.01 − 2.64i)8-s + (2.50 + 0.353i)10-s + (1.16 − 1.16i)11-s + (−0.842 − 0.842i)13-s + (1.67 − 1.25i)14-s + (−3.38 − 2.12i)16-s + 4.56i·17-s + (2.78 − 2.78i)19-s + (3.13 − 1.73i)20-s + (0.325 − 2.31i)22-s + 5.13i·23-s + ⋯
L(s)  = 1  + (0.798 − 0.601i)2-s + (0.276 − 0.961i)4-s + (0.566 + 0.566i)5-s + 0.558·7-s + (−0.357 − 0.933i)8-s + (0.792 + 0.111i)10-s + (0.352 − 0.352i)11-s + (−0.233 − 0.233i)13-s + (0.446 − 0.336i)14-s + (−0.847 − 0.531i)16-s + 1.10i·17-s + (0.639 − 0.639i)19-s + (0.700 − 0.387i)20-s + (0.0694 − 0.492i)22-s + 1.07i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.579 + 0.814i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14395 - 1.10616i\)
\(L(\frac12)\) \(\approx\) \(2.14395 - 1.10616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 + 0.850i)T \)
3 \( 1 \)
good5 \( 1 + (-1.26 - 1.26i)T + 5iT^{2} \)
7 \( 1 - 1.47T + 7T^{2} \)
11 \( 1 + (-1.16 + 1.16i)T - 11iT^{2} \)
13 \( 1 + (0.842 + 0.842i)T + 13iT^{2} \)
17 \( 1 - 4.56iT - 17T^{2} \)
19 \( 1 + (-2.78 + 2.78i)T - 19iT^{2} \)
23 \( 1 - 5.13iT - 23T^{2} \)
29 \( 1 + (0.161 - 0.161i)T - 29iT^{2} \)
31 \( 1 + 9.34iT - 31T^{2} \)
37 \( 1 + (6.53 - 6.53i)T - 37iT^{2} \)
41 \( 1 + 9.35T + 41T^{2} \)
43 \( 1 + (-3.98 - 3.98i)T + 43iT^{2} \)
47 \( 1 + 5.75T + 47T^{2} \)
53 \( 1 + (-7.87 - 7.87i)T + 53iT^{2} \)
59 \( 1 + (1.12 - 1.12i)T - 59iT^{2} \)
61 \( 1 + (-0.396 - 0.396i)T + 61iT^{2} \)
67 \( 1 + (-6.11 + 6.11i)T - 67iT^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 - 4.38iT - 79T^{2} \)
83 \( 1 + (4.10 + 4.10i)T + 83iT^{2} \)
89 \( 1 - 0.815T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−11.22319829818960984372240663742, −10.23482400102363908224609779824, −9.606431814092349258609791448417, −8.319437495182878296969362096986, −7.02457149059544438093080849427, −6.07005013965328035545623240062, −5.23410092558157358187999980152, −4.00067625956188780067340641434, −2.84524563398494547110956927308, −1.57209768420599461894811037409, 1.91358302429967757739507168100, 3.46314532643367148940041428184, 4.86298762413272513283720399209, 5.27497922765075633745750030071, 6.59532886863581842277700576095, 7.38798828872812825459982868618, 8.505334248128546123780472606669, 9.250168607320798929199641913346, 10.43941134982689458494132231494, 11.66902602291181636688964965558

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