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Properties

Label 2-18e2-27.22-c1-0-2
Degree $2$
Conductor $324$
Sign $-0.505 + 0.862i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
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  + (−2.29 + 0.836i)5-s + (−0.775 − 4.39i)7-s + (−2.73 − 0.996i)11-s + (−2.01 − 1.69i)13-s + (1.67 − 2.89i)17-s + (1.02 + 1.77i)19-s + (1.60 − 9.11i)23-s + (0.754 − 0.632i)25-s + (−5.30 + 4.45i)29-s + (−0.380 + 2.15i)31-s + (5.46 + 9.45i)35-s + (0.708 − 1.22i)37-s + (−3.13 − 2.63i)41-s + (4.42 + 1.61i)43-s + (1.03 + 5.89i)47-s + ⋯
L(s)  = 1  + (−1.02 + 0.374i)5-s + (−0.293 − 1.66i)7-s + (−0.825 − 0.300i)11-s + (−0.559 − 0.469i)13-s + (0.405 − 0.701i)17-s + (0.234 + 0.406i)19-s + (0.335 − 1.90i)23-s + (0.150 − 0.126i)25-s + (−0.985 + 0.826i)29-s + (−0.0683 + 0.387i)31-s + (0.923 + 1.59i)35-s + (0.116 − 0.201i)37-s + (−0.489 − 0.410i)41-s + (0.674 + 0.245i)43-s + (0.151 + 0.859i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.505 + 0.862i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.505 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326032 - 0.569073i\)
\(L(\frac12)\) \(\approx\) \(0.326032 - 0.569073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (2.29 - 0.836i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.775 + 4.39i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (2.73 + 0.996i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (2.01 + 1.69i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.02 - 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.60 + 9.11i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.30 - 4.45i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.380 - 2.15i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.13 + 2.63i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.42 - 1.61i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.03 - 5.89i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 1.97T + 53T^{2} \)
59 \( 1 + (-6.20 + 2.25i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.25 + 7.11i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.37 - 1.99i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-6.60 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.40 - 11.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.57 + 1.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.20 + 1.00i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.88 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.9 + 5.07i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.97609317267952153423980691169, −10.65104818671775099937944857637, −9.636533845376676346748058332200, −8.125943080893700008789364939743, −7.49183115588007251759238671906, −6.76918598465897994527022512560, −5.13053722867253383254968926491, −3.98561607199750286534173226674, −3.01758732012960419152755133305, −0.45395883519180593850241709693, 2.26996924729335978925051992326, 3.64525301439457991439154429527, 5.04250649024083962942455486829, 5.85410522520815102135689532635, 7.35732137451001850425340143688, 8.128466465635901435219070371660, 9.108490927153156578586781507352, 9.870924407671671626150046451295, 11.36652833837483219371062259224, 11.87683912502660570091842885958

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