Properties

Label 2-18e2-324.23-c1-0-26
Degree $2$
Conductor $324$
Sign $-0.288 - 0.957i$
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  + (1.09 + 0.891i)2-s + (0.890 + 1.48i)3-s + (0.411 + 1.95i)4-s + (−1.03 + 1.57i)5-s + (−0.345 + 2.42i)6-s + (3.59 − 3.39i)7-s + (−1.29 + 2.51i)8-s + (−1.41 + 2.64i)9-s + (−2.53 + 0.804i)10-s + (−1.99 − 1.00i)11-s + (−2.54 + 2.35i)12-s + (1.56 − 3.63i)13-s + (6.97 − 0.521i)14-s + (−3.25 − 0.136i)15-s + (−3.66 + 1.61i)16-s + (−0.421 + 0.502i)17-s + ⋯
L(s)  = 1  + (0.776 + 0.630i)2-s + (0.514 + 0.857i)3-s + (0.205 + 0.978i)4-s + (−0.461 + 0.702i)5-s + (−0.141 + 0.989i)6-s + (1.35 − 1.28i)7-s + (−0.456 + 0.889i)8-s + (−0.471 + 0.881i)9-s + (−0.800 + 0.254i)10-s + (−0.602 − 0.302i)11-s + (−0.733 + 0.679i)12-s + (0.435 − 1.00i)13-s + (1.86 − 0.139i)14-s + (−0.839 − 0.0351i)15-s + (−0.915 + 0.403i)16-s + (−0.102 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35485 + 1.82255i\)
\(L(\frac12)\) \(\approx\) \(1.35485 + 1.82255i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 - 0.891i)T \)
3 \( 1 + (-0.890 - 1.48i)T \)
good5 \( 1 + (1.03 - 1.57i)T + (-1.98 - 4.59i)T^{2} \)
7 \( 1 + (-3.59 + 3.39i)T + (0.407 - 6.98i)T^{2} \)
11 \( 1 + (1.99 + 1.00i)T + (6.56 + 8.82i)T^{2} \)
13 \( 1 + (-1.56 + 3.63i)T + (-8.92 - 9.45i)T^{2} \)
17 \( 1 + (0.421 - 0.502i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (1.94 + 2.32i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (0.969 - 1.02i)T + (-1.33 - 22.9i)T^{2} \)
29 \( 1 + (-0.764 + 6.54i)T + (-28.2 - 6.68i)T^{2} \)
31 \( 1 + (-0.256 + 1.08i)T + (-27.7 - 13.9i)T^{2} \)
37 \( 1 + (1.45 - 8.27i)T + (-34.7 - 12.6i)T^{2} \)
41 \( 1 + (-5.93 + 4.41i)T + (11.7 - 39.2i)T^{2} \)
43 \( 1 + (-6.22 + 0.362i)T + (42.7 - 4.99i)T^{2} \)
47 \( 1 + (-4.74 + 1.12i)T + (42.0 - 21.0i)T^{2} \)
53 \( 1 + (-5.22 - 3.01i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (13.0 - 6.53i)T + (35.2 - 47.3i)T^{2} \)
61 \( 1 + (3.33 - 11.1i)T + (-50.9 - 33.5i)T^{2} \)
67 \( 1 + (0.556 + 4.76i)T + (-65.1 + 15.4i)T^{2} \)
71 \( 1 + (-1.72 + 0.627i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.516 - 0.187i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.2 + 9.09i)T + (22.6 + 75.6i)T^{2} \)
83 \( 1 + (5.49 - 7.38i)T + (-23.8 - 79.5i)T^{2} \)
89 \( 1 + (-1.92 + 5.27i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-7.54 + 4.96i)T + (38.4 - 89.0i)T^{2} \)
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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

−11.68418144307648715494564494071, −10.84502663804298138878933868791, −10.47714632663447302062796865701, −8.700157419761948607791032630874, −7.82409570799237541797950227314, −7.41500299696722873756988240053, −5.79299178209891812508225270209, −4.64273014188192912591881670479, −3.91805469217483001592151400253, −2.77629048021570710495229977080, 1.56760041348633490548572367237, 2.50400192561245033981178850860, 4.18665912671863288952389468728, 5.18612842301668174785554522099, 6.22634479742880581410265636779, 7.61938212249766794227136423618, 8.622745900742956508382797506310, 9.183510577077946231266659232575, 10.87004885338269804980674344899, 11.64011568650721580787904665143

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