Properties

Label 2-432-16.5-c1-0-22
Degree $2$
Conductor $432$
Sign $0.997 - 0.0646i$
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  + (1.40 + 0.115i)2-s + (1.97 + 0.326i)4-s + (1.78 + 1.78i)5-s − 4.77i·7-s + (2.74 + 0.688i)8-s + (2.30 + 2.72i)10-s + (−1.61 − 1.61i)11-s + (−1.94 + 1.94i)13-s + (0.553 − 6.73i)14-s + (3.78 + 1.28i)16-s + 4.57·17-s + (−5.73 + 5.73i)19-s + (2.93 + 4.10i)20-s + (−2.09 − 2.46i)22-s − 0.0549i·23-s + ⋯
L(s)  = 1  + (0.996 + 0.0819i)2-s + (0.986 + 0.163i)4-s + (0.798 + 0.798i)5-s − 1.80i·7-s + (0.969 + 0.243i)8-s + (0.729 + 0.860i)10-s + (−0.487 − 0.487i)11-s + (−0.539 + 0.539i)13-s + (0.147 − 1.79i)14-s + (0.946 + 0.322i)16-s + 1.10·17-s + (−1.31 + 1.31i)19-s + (0.657 + 0.917i)20-s + (−0.445 − 0.525i)22-s − 0.0114i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.997 - 0.0646i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.997 - 0.0646i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77503 + 0.0897299i\)
\(L(\frac12)\) \(\approx\) \(2.77503 + 0.0897299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.115i)T \)
3 \( 1 \)
good5 \( 1 + (-1.78 - 1.78i)T + 5iT^{2} \)
7 \( 1 + 4.77iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.61i)T + 11iT^{2} \)
13 \( 1 + (1.94 - 1.94i)T - 13iT^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 + (5.73 - 5.73i)T - 19iT^{2} \)
23 \( 1 + 0.0549iT - 23T^{2} \)
29 \( 1 + (4.88 - 4.88i)T - 29iT^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 - 4.33iT - 41T^{2} \)
43 \( 1 + (1.74 + 1.74i)T + 43iT^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (2.91 + 2.91i)T + 53iT^{2} \)
59 \( 1 + (5.09 + 5.09i)T + 59iT^{2} \)
61 \( 1 + (-4.33 + 4.33i)T - 61iT^{2} \)
67 \( 1 + (2.89 - 2.89i)T - 67iT^{2} \)
71 \( 1 + 6.50iT - 71T^{2} \)
73 \( 1 + 9.76iT - 73T^{2} \)
79 \( 1 + 2.55T + 79T^{2} \)
83 \( 1 + (8.59 - 8.59i)T - 83iT^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + 6.93T + 97T^{2} \)
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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

−10.84729677972018195352735393047, −10.61421826367088641582958030325, −9.805719594793955045749699680921, −7.989642233248209185911463765945, −7.21187194431318641053116364932, −6.43603756586725891555000463014, −5.46860708481210396871896776419, −4.16235882668040016845212070931, −3.30377049303910674811892741423, −1.82332637866861874255794109732, 1.99835316597971446471144448648, 2.79794447696821550422356961121, 4.59680538610406668578612251534, 5.50875431312902365919476431846, 5.84928166353080911923699928460, 7.27684253734940080939161971587, 8.497592531639792315987609996349, 9.380248123660267530521865327538, 10.27436808544753362822236353732, 11.44470818166734520124354193733

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