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Properties

Label 2-432-16.13-c1-0-10
Degree $2$
Conductor $432$
Sign $0.923 + 0.382i$
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  − 1.41i·2-s − 2.00·4-s + (−0.707 + 0.707i)5-s + 3i·7-s + 2.82i·8-s + (1.00 + 1.00i)10-s + (3.53 − 3.53i)11-s + (1 + i)13-s + 4.24·14-s + 4.00·16-s + 4.24·17-s + (4 + 4i)19-s + (1.41 − 1.41i)20-s + (−5.00 − 5.00i)22-s − 2.82i·23-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (−0.316 + 0.316i)5-s + 1.13i·7-s + 1.00i·8-s + (0.316 + 0.316i)10-s + (1.06 − 1.06i)11-s + (0.277 + 0.277i)13-s + 1.13·14-s + 1.00·16-s + 1.02·17-s + (0.917 + 0.917i)19-s + (0.316 − 0.316i)20-s + (−1.06 − 1.06i)22-s − 0.589i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.923 + 0.382i$
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: \((2,\ 432,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23158 - 0.244976i\)
\(L(\frac12)\) \(\approx\) \(1.23158 - 0.244976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 + (0.707 - 0.707i)T - 5iT^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + (-3.53 + 3.53i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + (-4 - 4i)T + 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-2.82 - 2.82i)T + 29iT^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-7 + 7i)T - 43iT^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + (0.707 - 0.707i)T - 53iT^{2} \)
59 \( 1 + (7.07 - 7.07i)T - 59iT^{2} \)
61 \( 1 + (-10 - 10i)T + 61iT^{2} \)
67 \( 1 + (-1 - i)T + 67iT^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (7.77 + 7.77i)T + 83iT^{2} \)
89 \( 1 - 1.41iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−11.27505313208376439426094580464, −10.33844210618699093171829325537, −9.217444719297262353764642623054, −8.753116351751879270384504967628, −7.65763996633432484377989348785, −6.10129139749155499702450030233, −5.31126208466626089564556919339, −3.76483011059339621029102631114, −3.04983304176838091241052994574, −1.42832791712060923972908294814, 0.999190478304177383108947652588, 3.65042521964223451480747970573, 4.42636901150732962099004719497, 5.50197049362225323294404543938, 6.81752404760477237939586679878, 7.36904087970834219375644990118, 8.238299102829853821791758475115, 9.436784910739632970107072832850, 9.931005045558722894783029401260, 11.19576452969721465652139468441

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