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Properties

Label 2-432-48.35-c1-0-10
Degree $2$
Conductor $432$
Sign $0.993 + 0.112i$
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  + (−1.41 + 0.0987i)2-s + (1.98 − 0.278i)4-s + (0.0308 + 0.0308i)5-s + 2.10·7-s + (−2.76 + 0.588i)8-s + (−0.0465 − 0.0404i)10-s + (2.05 − 2.05i)11-s + (0.0744 + 0.0744i)13-s + (−2.97 + 0.208i)14-s + (3.84 − 1.10i)16-s + 1.91i·17-s + (0.131 − 0.131i)19-s + (0.0696 + 0.0524i)20-s + (−2.69 + 3.10i)22-s + 2.90i·23-s + ⋯
L(s)  = 1  + (−0.997 + 0.0698i)2-s + (0.990 − 0.139i)4-s + (0.0137 + 0.0137i)5-s + 0.796·7-s + (−0.978 + 0.208i)8-s + (−0.0147 − 0.0127i)10-s + (0.620 − 0.620i)11-s + (0.0206 + 0.0206i)13-s + (−0.794 + 0.0555i)14-s + (0.961 − 0.275i)16-s + 0.464i·17-s + (0.0302 − 0.0302i)19-s + (0.0155 + 0.0117i)20-s + (−0.575 + 0.661i)22-s + 0.604i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.993 + 0.112i$
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: \((2,\ 432,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02326 - 0.0579376i\)
\(L(\frac12)\) \(\approx\) \(1.02326 - 0.0579376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0987i)T \)
3 \( 1 \)
good5 \( 1 + (-0.0308 - 0.0308i)T + 5iT^{2} \)
7 \( 1 - 2.10T + 7T^{2} \)
11 \( 1 + (-2.05 + 2.05i)T - 11iT^{2} \)
13 \( 1 + (-0.0744 - 0.0744i)T + 13iT^{2} \)
17 \( 1 - 1.91iT - 17T^{2} \)
19 \( 1 + (-0.131 + 0.131i)T - 19iT^{2} \)
23 \( 1 - 2.90iT - 23T^{2} \)
29 \( 1 + (-6.63 + 6.63i)T - 29iT^{2} \)
31 \( 1 - 5.33iT - 31T^{2} \)
37 \( 1 + (-3.75 + 3.75i)T - 37iT^{2} \)
41 \( 1 - 2.42T + 41T^{2} \)
43 \( 1 + (-7.02 - 7.02i)T + 43iT^{2} \)
47 \( 1 - 9.23T + 47T^{2} \)
53 \( 1 + (-5.34 - 5.34i)T + 53iT^{2} \)
59 \( 1 + (-8.08 + 8.08i)T - 59iT^{2} \)
61 \( 1 + (9.29 + 9.29i)T + 61iT^{2} \)
67 \( 1 + (6.10 - 6.10i)T - 67iT^{2} \)
71 \( 1 - 8.70iT - 71T^{2} \)
73 \( 1 + 6.34iT - 73T^{2} \)
79 \( 1 - 6.95iT - 79T^{2} \)
83 \( 1 + (8.86 + 8.86i)T + 83iT^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 + 8.51T + 97T^{2} \)
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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

−11.05141502971537460018843046491, −10.22454390331750709702092509529, −9.232993070269738483639974496652, −8.397091111038951909407193564142, −7.72931675740718939804395962556, −6.55807516729276027731204611912, −5.71590218738893202740633556023, −4.20432995491376312072800817158, −2.63002821427877502805426932803, −1.15957476799989726746484468310, 1.28872837997109359677014098542, 2.64987159027829128221113556942, 4.25351049391575303245049961444, 5.59070801719488857091854257447, 6.84001484130649983399284086445, 7.54087142816253305575408746311, 8.577028010086392341541572542213, 9.271462258425682976231951027463, 10.24322381806020664042955968456, 11.05318603718904434748011206659

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