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Properties

Label 2-432-48.11-c1-0-14
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: $\chi_{432} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323850 - 0.249797i\)
\(L(\frac12)\) \(\approx\) \(0.323850 - 0.249797i\)
\(L(\frac{3}{2})\) 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

−10.86460337768098012609680048424, −9.914924124821336192123196978042, −9.290323154225453776032894104352, −8.126778952719255502759105838575, −7.26237170786900344287735744954, −6.55183253708895719427036586552, −5.64354632343276898187662636597, −3.67620630887804805518162064701, −2.75109031720127518457893117999, −0.36876785090231344550884535963, 1.41789174716330834096314051151, 3.39010936514601265528603072688, 4.00358384990438346230570539774, 6.02194459836264992604068052499, 6.77559811109626175141850152237, 7.904037246775574380787764700098, 8.953144562042063013033601702073, 9.201825935374919462882747430990, 10.42185107145583193689223947839, 11.27773677242819518963221001237

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