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Properties

Label 2-432-48.11-c1-0-19
Degree $2$
Conductor $432$
Sign $0.870 + 0.492i$
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  + (1.19 − 0.749i)2-s + (0.876 − 1.79i)4-s + (−2.15 + 2.15i)5-s + 4.43·7-s + (−0.295 − 2.81i)8-s + (−0.969 + 4.19i)10-s + (3.87 + 3.87i)11-s + (0.958 − 0.958i)13-s + (5.32 − 3.32i)14-s + (−2.46 − 3.15i)16-s − 2.92i·17-s + (−4.16 − 4.16i)19-s + (1.98 + 5.76i)20-s + (7.55 + 1.74i)22-s − 1.03i·23-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.963 + 0.963i)5-s + 1.67·7-s + (−0.104 − 0.994i)8-s + (−0.306 + 1.32i)10-s + (1.16 + 1.16i)11-s + (0.265 − 0.265i)13-s + (1.42 − 0.888i)14-s + (−0.615 − 0.788i)16-s − 0.709i·17-s + (−0.955 − 0.955i)19-s + (0.443 + 1.28i)20-s + (1.61 + 0.371i)22-s − 0.216i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.870 + 0.492i$
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: \((2,\ 432,\ (\ :1/2),\ 0.870 + 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22501 - 0.585882i\)
\(L(\frac12)\) \(\approx\) \(2.22501 - 0.585882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.749i)T \)
3 \( 1 \)
good5 \( 1 + (2.15 - 2.15i)T - 5iT^{2} \)
7 \( 1 - 4.43T + 7T^{2} \)
11 \( 1 + (-3.87 - 3.87i)T + 11iT^{2} \)
13 \( 1 + (-0.958 + 0.958i)T - 13iT^{2} \)
17 \( 1 + 2.92iT - 17T^{2} \)
19 \( 1 + (4.16 + 4.16i)T + 19iT^{2} \)
23 \( 1 + 1.03iT - 23T^{2} \)
29 \( 1 + (-0.941 - 0.941i)T + 29iT^{2} \)
31 \( 1 - 0.537iT - 31T^{2} \)
37 \( 1 + (-5.89 - 5.89i)T + 37iT^{2} \)
41 \( 1 + 3.55T + 41T^{2} \)
43 \( 1 + (8.23 - 8.23i)T - 43iT^{2} \)
47 \( 1 + 0.595T + 47T^{2} \)
53 \( 1 + (6.92 - 6.92i)T - 53iT^{2} \)
59 \( 1 + (7.50 + 7.50i)T + 59iT^{2} \)
61 \( 1 + (-0.900 + 0.900i)T - 61iT^{2} \)
67 \( 1 + (2.68 + 2.68i)T + 67iT^{2} \)
71 \( 1 + 0.430iT - 71T^{2} \)
73 \( 1 + 5.99iT - 73T^{2} \)
79 \( 1 + 0.295iT - 79T^{2} \)
83 \( 1 + (1.58 - 1.58i)T - 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + 3.49T + 97T^{2} \)
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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

−11.34149599078812212731102987420, −10.65115193152975628651199601545, −9.487963152554012130298772020254, −8.174825309979198398103377341241, −7.18177202485744188659226068903, −6.45981118601708217609064734109, −4.76576541305747057198021215583, −4.37578785011651262115295062079, −3.01351589137679066255748803266, −1.64190657526777534290145271405, 1.60570824745601472186617208319, 3.80749707767729630890611310323, 4.28901380055833154787134138471, 5.36807005752022205128567653467, 6.36179403661209192336050242726, 7.72447628732246477484368060679, 8.404720879608856209413025437281, 8.756507007248140121377111142495, 10.81050623233355652982114159785, 11.58742438234795728168394096434

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