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Properties

Label 2-432-48.35-c1-0-11
Degree $2$
Conductor $432$
Sign $0.253 - 0.967i$
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  + (1.32 + 0.496i)2-s + (1.50 + 1.31i)4-s + (1.75 + 1.75i)5-s − 4.05·7-s + (1.34 + 2.48i)8-s + (1.45 + 3.20i)10-s + (−1.00 + 1.00i)11-s + (4.19 + 4.19i)13-s + (−5.37 − 2.01i)14-s + (0.542 + 3.96i)16-s − 5.82i·17-s + (0.687 − 0.687i)19-s + (0.337 + 4.96i)20-s + (−1.82 + 0.828i)22-s − 3.75i·23-s + ⋯
L(s)  = 1  + (0.936 + 0.351i)2-s + (0.753 + 0.657i)4-s + (0.786 + 0.786i)5-s − 1.53·7-s + (0.474 + 0.880i)8-s + (0.460 + 1.01i)10-s + (−0.301 + 0.301i)11-s + (1.16 + 1.16i)13-s + (−1.43 − 0.538i)14-s + (0.135 + 0.990i)16-s − 1.41i·17-s + (0.157 − 0.157i)19-s + (0.0755 + 1.10i)20-s + (−0.388 + 0.176i)22-s − 0.783i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.253 - 0.967i$
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: \((2,\ 432,\ (\ :1/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90664 + 1.47066i\)
\(L(\frac12)\) \(\approx\) \(1.90664 + 1.47066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.496i)T \)
3 \( 1 \)
good5 \( 1 + (-1.75 - 1.75i)T + 5iT^{2} \)
7 \( 1 + 4.05T + 7T^{2} \)
11 \( 1 + (1.00 - 1.00i)T - 11iT^{2} \)
13 \( 1 + (-4.19 - 4.19i)T + 13iT^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 + (-0.687 + 0.687i)T - 19iT^{2} \)
23 \( 1 + 3.75iT - 23T^{2} \)
29 \( 1 + (-6.60 + 6.60i)T - 29iT^{2} \)
31 \( 1 - 0.621iT - 31T^{2} \)
37 \( 1 + (3.34 - 3.34i)T - 37iT^{2} \)
41 \( 1 + 7.83T + 41T^{2} \)
43 \( 1 + (-1.15 - 1.15i)T + 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (5.59 + 5.59i)T + 53iT^{2} \)
59 \( 1 + (-3.51 + 3.51i)T - 59iT^{2} \)
61 \( 1 + (-1.61 - 1.61i)T + 61iT^{2} \)
67 \( 1 + (3.84 - 3.84i)T - 67iT^{2} \)
71 \( 1 - 5.98iT - 71T^{2} \)
73 \( 1 + 5.11iT - 73T^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 + (7.07 + 7.07i)T + 83iT^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 + 8.67T + 97T^{2} \)
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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

−11.53052025416145988361256606813, −10.43695469423654085859941296022, −9.701513266313772904697941440009, −8.606003148288609466038437846319, −7.04561132555186057765759480779, −6.59172445860689769038184472946, −5.90492322466154359882961140941, −4.52154322236848780193266282707, −3.25945591859529055513464391450, −2.41573926242076432747583013133, 1.30180420171448107043774369438, 3.01346085257798565199475805054, 3.82872297645302007191886234285, 5.46270864017730083508292762233, 5.86616904270761245630417849317, 6.82046288602759696316664548951, 8.351208527081160228526179466272, 9.359925936416697964423620560689, 10.31208179043916740172029940385, 10.78302250191778209377506400179

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