Properties

Label 2-18e2-81.65-c2-0-1
Degree $2$
Conductor $324$
Sign $-0.178 - 0.983i$
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  + (−1.59 − 2.54i)3-s + (−2.52 − 1.87i)5-s + (−4.60 − 3.03i)7-s + (−3.93 + 8.09i)9-s + (0.644 + 5.51i)11-s + (3.12 − 10.4i)13-s + (−0.764 + 9.40i)15-s + (−9.72 + 1.71i)17-s + (−5.56 + 31.5i)19-s + (−0.379 + 16.5i)21-s + (11.1 + 16.9i)23-s + (−4.33 − 14.4i)25-s + (26.8 − 2.85i)27-s + (18.8 + 17.7i)29-s + (1.17 + 20.1i)31-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)3-s + (−0.504 − 0.375i)5-s + (−0.658 − 0.432i)7-s + (−0.437 + 0.899i)9-s + (0.0586 + 0.501i)11-s + (0.240 − 0.803i)13-s + (−0.0509 + 0.626i)15-s + (−0.572 + 0.100i)17-s + (−0.292 + 1.66i)19-s + (−0.0180 + 0.787i)21-s + (0.485 + 0.738i)23-s + (−0.173 − 0.578i)25-s + (0.994 − 0.105i)27-s + (0.649 + 0.612i)29-s + (0.0378 + 0.649i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.178 - 0.983i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.178 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.157996 + 0.189310i\)
\(L(\frac12)\) \(\approx\) \(0.157996 + 0.189310i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.59 + 2.54i)T \)
good5 \( 1 + (2.52 + 1.87i)T + (7.17 + 23.9i)T^{2} \)
7 \( 1 + (4.60 + 3.03i)T + (19.4 + 44.9i)T^{2} \)
11 \( 1 + (-0.644 - 5.51i)T + (-117. + 27.9i)T^{2} \)
13 \( 1 + (-3.12 + 10.4i)T + (-141. - 92.8i)T^{2} \)
17 \( 1 + (9.72 - 1.71i)T + (271. - 98.8i)T^{2} \)
19 \( 1 + (5.56 - 31.5i)T + (-339. - 123. i)T^{2} \)
23 \( 1 + (-11.1 - 16.9i)T + (-209. + 485. i)T^{2} \)
29 \( 1 + (-18.8 - 17.7i)T + (48.8 + 839. i)T^{2} \)
31 \( 1 + (-1.17 - 20.1i)T + (-954. + 111. i)T^{2} \)
37 \( 1 + (51.6 + 18.7i)T + (1.04e3 + 879. i)T^{2} \)
41 \( 1 + (-14.2 - 60.0i)T + (-1.50e3 + 754. i)T^{2} \)
43 \( 1 + (8.52 - 19.7i)T + (-1.26e3 - 1.34e3i)T^{2} \)
47 \( 1 + (52.0 + 3.03i)T + (2.19e3 + 256. i)T^{2} \)
53 \( 1 + (8.55 - 4.94i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (9.04 - 77.3i)T + (-3.38e3 - 802. i)T^{2} \)
61 \( 1 + (-19.5 + 9.84i)T + (2.22e3 - 2.98e3i)T^{2} \)
67 \( 1 + (42.3 + 44.8i)T + (-261. + 4.48e3i)T^{2} \)
71 \( 1 + (61.2 + 72.9i)T + (-875. + 4.96e3i)T^{2} \)
73 \( 1 + (46.6 + 39.1i)T + (925. + 5.24e3i)T^{2} \)
79 \( 1 + (-10.3 - 2.44i)T + (5.57e3 + 2.80e3i)T^{2} \)
83 \( 1 + (18.8 - 79.3i)T + (-6.15e3 - 3.09e3i)T^{2} \)
89 \( 1 + (-15.4 + 18.4i)T + (-1.37e3 - 7.80e3i)T^{2} \)
97 \( 1 + (91.4 + 122. i)T + (-2.69e3 + 9.01e3i)T^{2} \)
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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

−11.88089702560392196129000970113, −10.75094326006504366540634161972, −10.01400986842519823912818330649, −8.571364722423323200826170102592, −7.78580829438885105553681398083, −6.83067205697509633113394998848, −5.91258908014358230481450766670, −4.69024054532194608893594207317, −3.29493182749962378571940844812, −1.46445313582843789586466398939, 0.12716726392059312264689252830, 2.80702444287382931873507704314, 3.95305273409959343927174250254, 5.01583035023297748923487299591, 6.29996212539666327470177948603, 6.96348908861914216618529934094, 8.677520826735629938321019074019, 9.203715200920925871753194819168, 10.32490108622356195717002122679, 11.24922034015488120545887384243

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