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Properties

Label 2-432-432.133-c1-0-25
Degree $2$
Conductor $432$
Sign $-0.554 - 0.832i$
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  + (1.09 + 0.894i)2-s + (−0.896 + 1.48i)3-s + (0.399 + 1.95i)4-s + (2.61 + 0.228i)5-s + (−2.30 + 0.821i)6-s + (−0.721 − 1.98i)7-s + (−1.31 + 2.50i)8-s + (−1.39 − 2.65i)9-s + (2.65 + 2.58i)10-s + (0.572 + 6.54i)11-s + (−3.26 − 1.16i)12-s + (4.15 + 2.91i)13-s + (0.982 − 2.81i)14-s + (−2.67 + 3.66i)15-s + (−3.68 + 1.56i)16-s + (−2.55 − 4.41i)17-s + ⋯
L(s)  = 1  + (0.774 + 0.632i)2-s + (−0.517 + 0.855i)3-s + (0.199 + 0.979i)4-s + (1.16 + 0.102i)5-s + (−0.942 + 0.335i)6-s + (−0.272 − 0.749i)7-s + (−0.464 + 0.885i)8-s + (−0.464 − 0.885i)9-s + (0.839 + 0.817i)10-s + (0.172 + 1.97i)11-s + (−0.941 − 0.336i)12-s + (1.15 + 0.807i)13-s + (0.262 − 0.752i)14-s + (−0.691 + 0.946i)15-s + (−0.920 + 0.391i)16-s + (−0.618 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.957960 + 1.79010i\)
\(L(\frac12)\) \(\approx\) \(0.957960 + 1.79010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.894i)T \)
3 \( 1 + (0.896 - 1.48i)T \)
good5 \( 1 + (-2.61 - 0.228i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (0.721 + 1.98i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-0.572 - 6.54i)T + (-10.8 + 1.91i)T^{2} \)
13 \( 1 + (-4.15 - 2.91i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (2.55 + 4.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.40 + 5.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.746 - 2.05i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (2.46 + 3.52i)T + (-9.91 + 27.2i)T^{2} \)
31 \( 1 + (0.813 + 0.296i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-0.460 + 1.71i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.72 + 1.53i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.201 - 2.30i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-4.66 + 1.69i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-7.59 - 7.59i)T + 53iT^{2} \)
59 \( 1 + (-2.31 - 0.202i)T + (58.1 + 10.2i)T^{2} \)
61 \( 1 + (0.773 - 1.65i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (10.0 + 7.00i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-9.29 + 5.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.25 + 1.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.280 - 1.59i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.67 - 2.39i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-11.1 - 6.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.0173 + 0.0145i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.44923305230402427910320602141, −10.62876496519958757516228956224, −9.471953834028090068465280716773, −9.146803966688961328018428897080, −7.26700200152627798773241855226, −6.65528933917800980166066115813, −5.74860925299329574084475066271, −4.62048156428288546839190272635, −4.04585531411489748474057407785, −2.34158351254964214768743439386, 1.19393024602398487708794857520, 2.40225879441430634644411404135, 3.65479666392644052173298661765, 5.65687671766831358244644676131, 5.82309667368843502600837105525, 6.39960381503341432381177924012, 8.296451218367229199959237074068, 9.000923415562448987223764084530, 10.47092250875930756869444407394, 10.85846862476986111355008964040

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