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Properties

Label 2-18e2-9.4-c7-0-8
Degree $2$
Conductor $324$
Sign $0.984 - 0.173i$
Analytic cond. $101.212$
Root an. cond. $10.0604$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (103. − 178. i)5-s + (−503. − 872. i)7-s + (−1.65e3 − 2.86e3i)11-s + (−6.64e3 + 1.15e4i)13-s − 1.80e4·17-s + 2.17e4·19-s + (−3.24e4 + 5.61e4i)23-s + (1.77e4 + 3.07e4i)25-s + (1.05e5 + 1.83e5i)29-s + (−4.23e4 + 7.33e4i)31-s − 2.07e5·35-s + 1.63e5·37-s + (2.70e5 − 4.68e5i)41-s + (1.52e5 + 2.64e5i)43-s + (−4.05e5 − 7.01e5i)47-s + ⋯
L(s)  = 1  + (0.369 − 0.639i)5-s + (−0.554 − 0.960i)7-s + (−0.374 − 0.648i)11-s + (−0.838 + 1.45i)13-s − 0.888·17-s + 0.726·19-s + (−0.555 + 0.962i)23-s + (0.227 + 0.393i)25-s + (0.804 + 1.39i)29-s + (−0.255 + 0.442i)31-s − 0.819·35-s + 0.530·37-s + (0.613 − 1.06i)41-s + (0.292 + 0.507i)43-s + (−0.569 − 0.985i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(101.212\)
Root analytic conductor: \(10.0604\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :7/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.583799995\)
\(L(\frac12)\) \(\approx\) \(1.583799995\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-103. + 178. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (503. + 872. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (1.65e3 + 2.86e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (6.64e3 - 1.15e4i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 1.80e4T + 4.10e8T^{2} \)
19 \( 1 - 2.17e4T + 8.93e8T^{2} \)
23 \( 1 + (3.24e4 - 5.61e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-1.05e5 - 1.83e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (4.23e4 - 7.33e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 1.63e5T + 9.49e10T^{2} \)
41 \( 1 + (-2.70e5 + 4.68e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-1.52e5 - 2.64e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (4.05e5 + 7.01e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 - 1.40e6T + 1.17e12T^{2} \)
59 \( 1 + (-6.98e4 + 1.20e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.45e5 + 2.52e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-8.08e5 + 1.39e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 9.62e5T + 9.09e12T^{2} \)
73 \( 1 - 2.11e5T + 1.10e13T^{2} \)
79 \( 1 + (3.76e6 + 6.51e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.74e6 + 3.01e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 - 9.64e5T + 4.42e13T^{2} \)
97 \( 1 + (4.91e5 + 8.52e5i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−10.35051315736074876287548439538, −9.433451423316261847568685559166, −8.778286912255732120809527846522, −7.40260030150168967342418083876, −6.71922318993308482142302934987, −5.43824734475955922236197757208, −4.46858093027361627026013697665, −3.35814156849668297834877138458, −1.90998003373733242365255967884, −0.73371814429417731805103558697, 0.47325889413845179183703591014, 2.48903740952949283110673523067, 2.69749479621168479238345875466, 4.45182689549615654767750230387, 5.62644759708394512464394338247, 6.40097173622491938576037382112, 7.51576989449903818583881628338, 8.470288977555775699556599008964, 9.759535781622857236856674336684, 10.10886280946756974515012151865

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