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Properties

Label 2-18e2-9.2-c2-0-5
Degree $2$
Conductor $324$
Sign $0.573 + 0.819i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.01 − 2.89i)5-s + (3.09 − 5.36i)7-s + (−0.984 − 0.568i)11-s + (5.69 + 9.86i)13-s − 31.2i·17-s − 32.9·19-s + (29.1 − 16.8i)23-s + (4.29 − 7.43i)25-s + (22.1 + 12.7i)29-s + (11.5 + 20.0i)31-s − 35.9i·35-s + 8.80·37-s + (61.2 − 35.3i)41-s + (29.8 − 51.7i)43-s + (−52.1 − 30.1i)47-s + ⋯
L(s)  = 1  + (1.00 − 0.579i)5-s + (0.442 − 0.766i)7-s + (−0.0895 − 0.0516i)11-s + (0.438 + 0.758i)13-s − 1.83i·17-s − 1.73·19-s + (1.26 − 0.731i)23-s + (0.171 − 0.297i)25-s + (0.763 + 0.440i)29-s + (0.373 + 0.647i)31-s − 1.02i·35-s + 0.237·37-s + (1.49 − 0.861i)41-s + (0.694 − 1.20i)43-s + (−1.10 − 0.640i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.573 + 0.819i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.76072 - 0.916575i\)
\(L(\frac12)\) \(\approx\) \(1.76072 - 0.916575i\)
\(L(2)\) 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 \( 1 + (-5.01 + 2.89i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-3.09 + 5.36i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.984 + 0.568i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-5.69 - 9.86i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 31.2iT - 289T^{2} \)
19 \( 1 + 32.9T + 361T^{2} \)
23 \( 1 + (-29.1 + 16.8i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.1 - 12.7i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.5 - 20.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 8.80T + 1.36e3T^{2} \)
41 \( 1 + (-61.2 + 35.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-29.8 + 51.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (52.1 + 30.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.7iT - 2.80e3T^{2} \)
59 \( 1 + (72.2 - 41.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.3 - 26.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.29 - 9.16i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 3.63iT - 5.04e3T^{2} \)
73 \( 1 + 17.2T + 5.32e3T^{2} \)
79 \( 1 + (56.6 - 98.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (10.2 + 5.90i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + (37.1 - 64.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.06857016006503301956933164243, −10.42158979429380715303015213986, −9.223166637188818007642414927578, −8.709426111935592275643324119614, −7.28924490134310471858295518966, −6.42028702793774520122934744113, −5.12692177966314860392082993104, −4.31753499944601869780977102086, −2.48800647093926439036644607934, −1.03009884883773354930612710232, 1.74831102416278600532033768098, 2.89582744796415355141147947983, 4.51441574109500847156369189227, 5.99241964669778469109354465404, 6.23241727568573466398738359433, 7.908599026640170717436353739448, 8.672098106191475610643161833042, 9.775881277206372327194370805380, 10.65722940274643838654753232319, 11.27473159321755285161606356560

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