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Properties

Label 2-18e2-9.4-c5-0-9
Degree $2$
Conductor $324$
Sign $0.766 - 0.642i$
Analytic cond. $51.9643$
Root an. cond. $7.20863$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (27 − 46.7i)5-s + (44 + 76.2i)7-s + (270 + 467. i)11-s + (209 − 361. i)13-s − 594·17-s + 836·19-s + (−2.05e3 + 3.55e3i)23-s + (104.5 + 180. i)25-s + (−297 − 514. i)29-s + (−2.12e3 + 3.68e3i)31-s + 4.75e3·35-s − 298·37-s + (8.61e3 − 1.49e4i)41-s + (6.05e3 + 1.04e4i)43-s + (−648 − 1.12e3i)47-s + ⋯
L(s)  = 1  + (0.482 − 0.836i)5-s + (0.339 + 0.587i)7-s + (0.672 + 1.16i)11-s + (0.342 − 0.594i)13-s − 0.498·17-s + 0.531·19-s + (−0.808 + 1.40i)23-s + (0.0334 + 0.0579i)25-s + (−0.0655 − 0.113i)29-s + (−0.397 + 0.688i)31-s + 0.655·35-s − 0.0357·37-s + (0.800 − 1.38i)41-s + (0.498 + 0.864i)43-s + (−0.0427 − 0.0741i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(51.9643\)
Root analytic conductor: \(7.20863\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :5/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.367483430\)
\(L(\frac12)\) \(\approx\) \(2.367483430\)
\(L(\frac{7}{2})\) 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 \( 1 + (-27 + 46.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-44 - 76.2i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-270 - 467. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-209 + 361. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 594T + 1.41e6T^{2} \)
19 \( 1 - 836T + 2.47e6T^{2} \)
23 \( 1 + (2.05e3 - 3.55e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (297 + 514. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.12e3 - 3.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 298T + 6.93e7T^{2} \)
41 \( 1 + (-8.61e3 + 1.49e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-6.05e3 - 1.04e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (648 + 1.12e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 1.94e4T + 4.18e8T^{2} \)
59 \( 1 + (3.83e3 - 6.64e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.73e4 - 3.00e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.09e4 - 1.88e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.68e4T + 1.80e9T^{2} \)
73 \( 1 - 6.75e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.84e4 - 6.66e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.38e4 - 5.86e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 2.97e4T + 5.58e9T^{2} \)
97 \( 1 + (-6.11e4 - 1.05e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−10.92649702607682311703254955277, −9.655934411275165945831790647707, −9.187074444656611724322628695851, −8.139075808496025515233764145686, −7.07360021684453199247855318453, −5.76077477337314444184524114433, −5.04225698676996077440506787299, −3.82542783912059056199759849929, −2.14928993905842677493514034177, −1.17859324885779690518017578175, 0.68763713007213654333199162349, 2.12434343778050494847770796435, 3.42644155167109205238045327849, 4.50405718249056239814185247069, 6.07879253362114707887642589915, 6.58951439953734889794889125230, 7.80154171127378889501329440786, 8.828681445213981244861034284480, 9.807943831200822130288449857749, 10.89887011771914365320340156175

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