Properties

Label 2-18e2-108.31-c2-0-13
Degree $2$
Conductor $324$
Sign $0.470 - 0.882i$
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  + (−0.747 + 1.85i)2-s + (−2.88 − 2.77i)4-s + (−0.463 + 2.62i)5-s + (−4.34 − 5.18i)7-s + (7.29 − 3.27i)8-s + (−4.52 − 2.82i)10-s + (18.5 − 3.26i)11-s + (−4.76 + 1.73i)13-s + (12.8 − 4.19i)14-s + (0.616 + 15.9i)16-s + (−7.37 + 12.7i)17-s + (28.7 − 16.6i)19-s + (8.62 − 6.28i)20-s + (−7.79 + 36.8i)22-s + (−11.3 + 13.5i)23-s + ⋯
L(s)  = 1  + (−0.373 + 0.927i)2-s + (−0.720 − 0.693i)4-s + (−0.0926 + 0.525i)5-s + (−0.621 − 0.740i)7-s + (0.912 − 0.409i)8-s + (−0.452 − 0.282i)10-s + (1.68 − 0.297i)11-s + (−0.366 + 0.133i)13-s + (0.918 − 0.299i)14-s + (0.0385 + 0.999i)16-s + (−0.433 + 0.751i)17-s + (1.51 − 0.875i)19-s + (0.431 − 0.314i)20-s + (−0.354 + 1.67i)22-s + (−0.493 + 0.588i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.470 - 0.882i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.470 - 0.882i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07259 + 0.643592i\)
\(L(\frac12)\) \(\approx\) \(1.07259 + 0.643592i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.747 - 1.85i)T \)
3 \( 1 \)
good5 \( 1 + (0.463 - 2.62i)T + (-23.4 - 8.55i)T^{2} \)
7 \( 1 + (4.34 + 5.18i)T + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (-18.5 + 3.26i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (4.76 - 1.73i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (7.37 - 12.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-28.7 + 16.6i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.3 - 13.5i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-38.2 - 13.9i)T + (644. + 540. i)T^{2} \)
31 \( 1 + (4.43 - 5.28i)T + (-166. - 946. i)T^{2} \)
37 \( 1 + (4.65 - 8.05i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-20.9 + 7.61i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-42.0 + 7.40i)T + (1.73e3 - 632. i)T^{2} \)
47 \( 1 + (-35.0 - 41.7i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 + 25.6T + 2.80e3T^{2} \)
59 \( 1 + (-15.3 - 2.70i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (48.6 - 40.8i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (11.8 + 32.5i)T + (-3.43e3 + 2.88e3i)T^{2} \)
71 \( 1 + (-23.7 - 13.6i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-18.2 - 31.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-43.9 + 120. i)T + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (0.482 - 1.32i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-34.1 - 59.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (19.7 + 111. i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−11.36696889334159218657983580380, −10.43155724077867443728726327798, −9.489466506504008390952045273518, −8.829963902134017593475972321039, −7.42716392452572489819610720061, −6.84294592723123812431357656948, −6.02874580917800539474974785748, −4.53588177080610985477504249071, −3.43226163967387392971521125741, −1.04490562182167736460827163508, 0.967616446356373577247515405175, 2.55449249792087346493946031238, 3.83590947288802099729986415449, 4.94446233012220162892045006950, 6.36376095977142663937799272772, 7.64182376555499422073119630663, 8.853019335512123779902271055371, 9.350896121495289733505249990431, 10.12576131126692504688400939516, 11.47562425252144703191215779767

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