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Properties

Label 2-432-48.11-c1-0-22
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: $\chi_{432} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66902 + 0.151120i\)
\(L(\frac12)\) \(\approx\) \(2.66902 + 0.151120i\)
\(L(\frac{3}{2})\) 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

−11.27192406263669145017838119416, −10.68628646447444324913011573490, −9.426438327865060617929195798904, −8.024992384688256300681344847743, −7.56378245985369003536214067344, −6.15495945744854274741110299597, −5.42392191236325929498112769394, −4.36477696867689281059170536010, −3.22908189852255033133077511529, −1.82650166665064227996460748284, 1.80218327877765817061759358390, 3.06918793249803896309589430979, 4.51461347776387306161231006764, 5.10165410658887484749566326069, 6.30088819753229681771366541584, 7.35162603628719563138720247778, 8.114086345745818093362230087149, 9.470822941717214070514744301547, 10.63098490053854304761723657856, 11.13757259742167793372532649106

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