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Properties

Label 2-432-48.11-c1-0-1
Degree $2$
Conductor $432$
Sign $-0.610 - 0.792i$
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  + (0.932 − 1.06i)2-s + (−0.262 − 1.98i)4-s + (−2.82 + 2.82i)5-s − 3.49·7-s + (−2.35 − 1.56i)8-s + (0.371 + 5.63i)10-s + (−1.04 − 1.04i)11-s + (0.232 − 0.232i)13-s + (−3.25 + 3.71i)14-s + (−3.86 + 1.04i)16-s + 6.83i·17-s + (−3.59 − 3.59i)19-s + (6.34 + 4.85i)20-s + (−2.08 + 0.137i)22-s + 3.02i·23-s + ⋯
L(s)  = 1  + (0.659 − 0.752i)2-s + (−0.131 − 0.991i)4-s + (−1.26 + 1.26i)5-s − 1.31·7-s + (−0.832 − 0.554i)8-s + (0.117 + 1.78i)10-s + (−0.314 − 0.314i)11-s + (0.0643 − 0.0643i)13-s + (−0.869 + 0.992i)14-s + (−0.965 + 0.260i)16-s + 1.65i·17-s + (−0.824 − 0.824i)19-s + (1.41 + 1.08i)20-s + (−0.444 + 0.0293i)22-s + 0.631i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.610 - 0.792i$
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: \((2,\ 432,\ (\ :1/2),\ -0.610 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0564918 + 0.114797i\)
\(L(\frac12)\) \(\approx\) \(0.0564918 + 0.114797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.932 + 1.06i)T \)
3 \( 1 \)
good5 \( 1 + (2.82 - 2.82i)T - 5iT^{2} \)
7 \( 1 + 3.49T + 7T^{2} \)
11 \( 1 + (1.04 + 1.04i)T + 11iT^{2} \)
13 \( 1 + (-0.232 + 0.232i)T - 13iT^{2} \)
17 \( 1 - 6.83iT - 17T^{2} \)
19 \( 1 + (3.59 + 3.59i)T + 19iT^{2} \)
23 \( 1 - 3.02iT - 23T^{2} \)
29 \( 1 + (4.56 + 4.56i)T + 29iT^{2} \)
31 \( 1 - 1.60iT - 31T^{2} \)
37 \( 1 + (-0.0478 - 0.0478i)T + 37iT^{2} \)
41 \( 1 - 5.73T + 41T^{2} \)
43 \( 1 + (-8.10 + 8.10i)T - 43iT^{2} \)
47 \( 1 + 4.46T + 47T^{2} \)
53 \( 1 + (3.08 - 3.08i)T - 53iT^{2} \)
59 \( 1 + (2.53 + 2.53i)T + 59iT^{2} \)
61 \( 1 + (5.63 - 5.63i)T - 61iT^{2} \)
67 \( 1 + (4.61 + 4.61i)T + 67iT^{2} \)
71 \( 1 + 5.63iT - 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 - 4.86iT - 79T^{2} \)
83 \( 1 + (0.847 - 0.847i)T - 83iT^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
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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

−11.31553082779126376702197039262, −10.78668187471812748346336006936, −10.09266310019168573864332444150, −8.928812753690322939026359969992, −7.63887684565442515908306115731, −6.58617295116698963999140437260, −5.93070167794499346742358896332, −4.13129099601178921261252489991, −3.52126861792246972042707703801, −2.56919266002792791357484060451, 0.06174439674123395236440989202, 3.04435956938314157445880549730, 4.12035673288933892808216731646, 4.89521935567946203892865204243, 6.04976539516441511080595438193, 7.18516941382227885064438911683, 7.87826711274022253536523287972, 8.888706657091705331090628253527, 9.552338895519417141231417645343, 11.16294390858336348644063652415

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