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Properties

Label 2-432-48.11-c1-0-15
Degree $2$
Conductor $432$
Sign $-0.587 + 0.809i$
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  + (−0.715 − 1.22i)2-s + (−0.976 + 1.74i)4-s + (−2.47 + 2.47i)5-s − 0.311·7-s + (2.82 − 0.0563i)8-s + (4.78 + 1.24i)10-s + (−3.17 − 3.17i)11-s + (3.97 − 3.97i)13-s + (0.222 + 0.379i)14-s + (−2.09 − 3.40i)16-s − 5.33i·17-s + (0.217 + 0.217i)19-s + (−1.90 − 6.73i)20-s + (−1.60 + 6.14i)22-s − 5.06i·23-s + ⋯
L(s)  = 1  + (−0.505 − 0.862i)2-s + (−0.488 + 0.872i)4-s + (−1.10 + 1.10i)5-s − 0.117·7-s + (0.999 − 0.0199i)8-s + (1.51 + 0.394i)10-s + (−0.956 − 0.956i)11-s + (1.10 − 1.10i)13-s + (0.0595 + 0.101i)14-s + (−0.522 − 0.852i)16-s − 1.29i·17-s + (0.0499 + 0.0499i)19-s + (−0.424 − 1.50i)20-s + (−0.341 + 1.30i)22-s − 1.05i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.587 + 0.809i$
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: \((2,\ 432,\ (\ :1/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255330 - 0.500862i\)
\(L(\frac12)\) \(\approx\) \(0.255330 - 0.500862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.715 + 1.22i)T \)
3 \( 1 \)
good5 \( 1 + (2.47 - 2.47i)T - 5iT^{2} \)
7 \( 1 + 0.311T + 7T^{2} \)
11 \( 1 + (3.17 + 3.17i)T + 11iT^{2} \)
13 \( 1 + (-3.97 + 3.97i)T - 13iT^{2} \)
17 \( 1 + 5.33iT - 17T^{2} \)
19 \( 1 + (-0.217 - 0.217i)T + 19iT^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + (-5.16 - 5.16i)T + 29iT^{2} \)
31 \( 1 + 4.95iT - 31T^{2} \)
37 \( 1 + (-1.46 - 1.46i)T + 37iT^{2} \)
41 \( 1 - 0.728T + 41T^{2} \)
43 \( 1 + (2.55 - 2.55i)T - 43iT^{2} \)
47 \( 1 - 7.10T + 47T^{2} \)
53 \( 1 + (2.78 - 2.78i)T - 53iT^{2} \)
59 \( 1 + (7.25 + 7.25i)T + 59iT^{2} \)
61 \( 1 + (8.33 - 8.33i)T - 61iT^{2} \)
67 \( 1 + (9.25 + 9.25i)T + 67iT^{2} \)
71 \( 1 + 4.78iT - 71T^{2} \)
73 \( 1 + 9.78iT - 73T^{2} \)
79 \( 1 + 15.2iT - 79T^{2} \)
83 \( 1 + (-0.0624 + 0.0624i)T - 83iT^{2} \)
89 \( 1 - 1.82T + 89T^{2} \)
97 \( 1 - 1.66T + 97T^{2} \)
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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

−10.74496233750886564627748292909, −10.43151901829303081475567693530, −9.001969201491357027642665066740, −8.048217641452619183741017790522, −7.55866510649017986885433159969, −6.25895860420597271679303026094, −4.68946355250588542480389780037, −3.23371536518580162234793989776, −2.94904735037521699407697624602, −0.45597391551035247786899179564, 1.46866445078727701486537654007, 3.99518034133191253212661479652, 4.71347906100872417593090305635, 5.86091876662402402106602480647, 7.02131063441904476143486143525, 7.975382877977946218527336519570, 8.513331139434769105477006058545, 9.384893385588033372349380786292, 10.41128826666435768895037578116, 11.40297963705974175724119190169

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