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Properties

Label 2-432-16.13-c1-0-20
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: $\chi_{432} (109, \cdot )$
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

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

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