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Properties

Label 2-432-16.13-c1-0-19
Degree $2$
Conductor $432$
Sign $-0.955 + 0.295i$
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  + (−1.15 − 0.812i)2-s + (0.678 + 1.88i)4-s + (−2.39 + 2.39i)5-s − 2.42i·7-s + (0.744 − 2.72i)8-s + (4.71 − 0.824i)10-s + (−0.803 + 0.803i)11-s + (0.643 + 0.643i)13-s + (−1.97 + 2.81i)14-s + (−3.07 + 2.55i)16-s − 0.717·17-s + (−3.76 − 3.76i)19-s + (−6.12 − 2.87i)20-s + (1.58 − 0.276i)22-s − 7.35i·23-s + ⋯
L(s)  = 1  + (−0.818 − 0.574i)2-s + (0.339 + 0.940i)4-s + (−1.07 + 1.07i)5-s − 0.918i·7-s + (0.263 − 0.964i)8-s + (1.49 − 0.260i)10-s + (−0.242 + 0.242i)11-s + (0.178 + 0.178i)13-s + (−0.527 + 0.751i)14-s + (−0.769 + 0.638i)16-s − 0.174·17-s + (−0.864 − 0.864i)19-s + (−1.37 − 0.643i)20-s + (0.337 − 0.0589i)22-s − 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.955 + 0.295i$
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: \((2,\ 432,\ (\ :1/2),\ -0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0310325 - 0.205696i\)
\(L(\frac12)\) \(\approx\) \(0.0310325 - 0.205696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.812i)T \)
3 \( 1 \)
good5 \( 1 + (2.39 - 2.39i)T - 5iT^{2} \)
7 \( 1 + 2.42iT - 7T^{2} \)
11 \( 1 + (0.803 - 0.803i)T - 11iT^{2} \)
13 \( 1 + (-0.643 - 0.643i)T + 13iT^{2} \)
17 \( 1 + 0.717T + 17T^{2} \)
19 \( 1 + (3.76 + 3.76i)T + 19iT^{2} \)
23 \( 1 + 7.35iT - 23T^{2} \)
29 \( 1 + (6.75 + 6.75i)T + 29iT^{2} \)
31 \( 1 + 6.69T + 31T^{2} \)
37 \( 1 + (7.02 - 7.02i)T - 37iT^{2} \)
41 \( 1 + 3.53iT - 41T^{2} \)
43 \( 1 + (-1.31 + 1.31i)T - 43iT^{2} \)
47 \( 1 - 4.02T + 47T^{2} \)
53 \( 1 + (6.85 - 6.85i)T - 53iT^{2} \)
59 \( 1 + (-4.81 + 4.81i)T - 59iT^{2} \)
61 \( 1 + (8.63 + 8.63i)T + 61iT^{2} \)
67 \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \)
71 \( 1 - 8.42iT - 71T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + (-6.14 - 6.14i)T + 83iT^{2} \)
89 \( 1 - 5.44iT - 89T^{2} \)
97 \( 1 + 7.77T + 97T^{2} \)
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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

−10.86641325866149562175904340050, −10.13005118516821273492667772526, −8.925131353559016900237852226781, −7.951548874248590532213775210855, −7.19924039613658723025269318319, −6.58296985310416486255447901094, −4.36389459100296423043776534039, −3.59894100243178529759751097675, −2.34527171688737838734055096867, −0.16741475413730415675664326726, 1.72148943624222253225272537398, 3.71704732308834093815785411818, 5.18119423227341639786248534029, 5.79045672673561355407377406239, 7.26799119218699490742768751206, 8.021767207274008311855032304912, 8.828990366878285445576836507951, 9.292830668984817681910199941245, 10.67703280756539688436146592110, 11.43915675169458033815466365908

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