Properties

Label 2-18e2-27.4-c1-0-2
Degree $2$
Conductor $324$
Sign $0.439 + 0.898i$
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  + (0.399 − 2.26i)5-s + (−0.715 + 0.600i)7-s + (−0.843 − 4.78i)11-s + (5.71 − 2.07i)13-s + (−1.42 + 2.47i)17-s + (−2.50 − 4.34i)19-s + (1.51 + 1.27i)23-s + (−0.277 − 0.101i)25-s + (−1.76 − 0.642i)29-s + (1.02 + 0.856i)31-s + (1.07 + 1.86i)35-s + (3.00 − 5.20i)37-s + (10.5 − 3.82i)41-s + (1.31 + 7.45i)43-s + (−8.37 + 7.02i)47-s + ⋯
L(s)  = 1  + (0.178 − 1.01i)5-s + (−0.270 + 0.226i)7-s + (−0.254 − 1.44i)11-s + (1.58 − 0.576i)13-s + (−0.346 + 0.600i)17-s + (−0.575 − 0.996i)19-s + (0.316 + 0.265i)23-s + (−0.0555 − 0.0202i)25-s + (−0.328 − 0.119i)29-s + (0.183 + 0.153i)31-s + (0.181 + 0.314i)35-s + (0.493 − 0.855i)37-s + (1.64 − 0.597i)41-s + (0.200 + 1.13i)43-s + (−1.22 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10099 - 0.686693i\)
\(L(\frac12)\) \(\approx\) \(1.10099 - 0.686693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.399 + 2.26i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.715 - 0.600i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.843 + 4.78i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-5.71 + 2.07i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.42 - 2.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.50 + 4.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.51 - 1.27i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.76 + 0.642i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-1.02 - 0.856i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.00 + 5.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.5 + 3.82i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.31 - 7.45i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.37 - 7.02i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 2.60T + 53T^{2} \)
59 \( 1 + (0.763 - 4.33i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (8.46 - 7.10i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.726 + 0.264i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.62 - 8.01i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.221 - 0.383i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.92 + 1.79i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-14.8 - 5.41i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-7.58 - 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.21 + 6.86i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.13902941715109556970741610206, −10.87152307660620623887570745823, −9.251843103148889624243291982239, −8.742076561510728484866365038644, −7.925852234168324433682122160922, −6.24631854691724879888369273111, −5.67428479181382997194125305002, −4.32207521359799515735586556489, −3.01284572565083281372582851308, −1.01870915041978836659019628592, 2.00124140937110846016071222003, 3.43959484275562931508161265792, 4.61489626032794567575042187974, 6.20876295016282818152658352983, 6.81467795436797300895775969195, 7.85951034713172424567497045920, 9.091962524403332927297454088917, 10.07391219705426966316501361525, 10.76193402165921852724957797458, 11.63882102922950410645545622469

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