Properties

Label 2-18e2-81.65-c2-0-12
Degree $2$
Conductor $324$
Sign $0.448 + 0.893i$
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  + (−0.463 + 2.96i)3-s + (−2.73 − 2.03i)5-s + (−1.12 − 0.737i)7-s + (−8.57 − 2.74i)9-s + (−0.325 − 2.78i)11-s + (6.15 − 20.5i)13-s + (7.29 − 7.15i)15-s + (5.48 − 0.967i)17-s + (0.824 − 4.67i)19-s + (2.70 − 2.98i)21-s + (10.9 + 16.6i)23-s + (−3.84 − 12.8i)25-s + (12.1 − 24.1i)27-s + (−14.1 − 13.3i)29-s + (−0.486 − 8.35i)31-s + ⋯
L(s)  = 1  + (−0.154 + 0.987i)3-s + (−0.546 − 0.406i)5-s + (−0.160 − 0.105i)7-s + (−0.952 − 0.305i)9-s + (−0.0295 − 0.253i)11-s + (0.473 − 1.58i)13-s + (0.486 − 0.476i)15-s + (0.322 − 0.0569i)17-s + (0.0434 − 0.246i)19-s + (0.128 − 0.142i)21-s + (0.475 + 0.722i)23-s + (−0.153 − 0.513i)25-s + (0.448 − 0.893i)27-s + (−0.488 − 0.460i)29-s + (−0.0157 − 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.448 + 0.893i$
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: \((2,\ 324,\ (\ :1),\ 0.448 + 0.893i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.830177 - 0.512104i\)
\(L(\frac12)\) \(\approx\) \(0.830177 - 0.512104i\)
\(L(2)\) 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 \( 1 + (0.463 - 2.96i)T \)
good5 \( 1 + (2.73 + 2.03i)T + (7.17 + 23.9i)T^{2} \)
7 \( 1 + (1.12 + 0.737i)T + (19.4 + 44.9i)T^{2} \)
11 \( 1 + (0.325 + 2.78i)T + (-117. + 27.9i)T^{2} \)
13 \( 1 + (-6.15 + 20.5i)T + (-141. - 92.8i)T^{2} \)
17 \( 1 + (-5.48 + 0.967i)T + (271. - 98.8i)T^{2} \)
19 \( 1 + (-0.824 + 4.67i)T + (-339. - 123. i)T^{2} \)
23 \( 1 + (-10.9 - 16.6i)T + (-209. + 485. i)T^{2} \)
29 \( 1 + (14.1 + 13.3i)T + (48.8 + 839. i)T^{2} \)
31 \( 1 + (0.486 + 8.35i)T + (-954. + 111. i)T^{2} \)
37 \( 1 + (-34.4 - 12.5i)T + (1.04e3 + 879. i)T^{2} \)
41 \( 1 + (6.96 + 29.3i)T + (-1.50e3 + 754. i)T^{2} \)
43 \( 1 + (-24.2 + 56.1i)T + (-1.26e3 - 1.34e3i)T^{2} \)
47 \( 1 + (-16.7 - 0.974i)T + (2.19e3 + 256. i)T^{2} \)
53 \( 1 + (84.4 - 48.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-4.20 + 35.9i)T + (-3.38e3 - 802. i)T^{2} \)
61 \( 1 + (-0.599 + 0.301i)T + (2.22e3 - 2.98e3i)T^{2} \)
67 \( 1 + (-23.3 - 24.7i)T + (-261. + 4.48e3i)T^{2} \)
71 \( 1 + (56.0 + 66.8i)T + (-875. + 4.96e3i)T^{2} \)
73 \( 1 + (70.3 + 59.0i)T + (925. + 5.24e3i)T^{2} \)
79 \( 1 + (31.3 + 7.43i)T + (5.57e3 + 2.80e3i)T^{2} \)
83 \( 1 + (12.1 - 51.0i)T + (-6.15e3 - 3.09e3i)T^{2} \)
89 \( 1 + (75.7 - 90.2i)T + (-1.37e3 - 7.80e3i)T^{2} \)
97 \( 1 + (34.4 + 46.2i)T + (-2.69e3 + 9.01e3i)T^{2} \)
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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

−11.10745144839024281693842863249, −10.36583035564951162279043675305, −9.431841433406656503920357034420, −8.455220735336213605740138457023, −7.64393415731145340843652768259, −6.02252274216736741018428862532, −5.19507347533180863151097996790, −4.00285819705863770051618540730, −3.05214155193005099919443361004, −0.48997824125248561224375188306, 1.53017091865621372183395886371, 3.00890282513474372706223271192, 4.43006257386518891393206730509, 5.93922543073449944699728379353, 6.81321067522199663752681296571, 7.56964764853530904931236081570, 8.604238422750257508234844932197, 9.586909976646937723051393862216, 11.07916458131745732568179315543, 11.45940708549477697577833448238

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