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Properties

Label 2-18e2-9.5-c2-0-6
Degree $2$
Conductor $324$
Sign $-0.573 + 0.819i$
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  + (−5.01 − 2.89i)5-s + (3.09 + 5.36i)7-s + (0.984 − 0.568i)11-s + (5.69 − 9.86i)13-s − 31.2i·17-s − 32.9·19-s + (−29.1 − 16.8i)23-s + (4.29 + 7.43i)25-s + (−22.1 + 12.7i)29-s + (11.5 − 20.0i)31-s − 35.9i·35-s + 8.80·37-s + (−61.2 − 35.3i)41-s + (29.8 + 51.7i)43-s + (52.1 − 30.1i)47-s + ⋯
L(s)  = 1  + (−1.00 − 0.579i)5-s + (0.442 + 0.766i)7-s + (0.0895 − 0.0516i)11-s + (0.438 − 0.758i)13-s − 1.83i·17-s − 1.73·19-s + (−1.26 − 0.731i)23-s + (0.171 + 0.297i)25-s + (−0.763 + 0.440i)29-s + (0.373 − 0.647i)31-s − 1.02i·35-s + 0.237·37-s + (−1.49 − 0.861i)41-s + (0.694 + 1.20i)43-s + (1.10 − 0.640i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.573 + 0.819i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.573 + 0.819i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.359133 - 0.689888i\)
\(L(\frac12)\) \(\approx\) \(0.359133 - 0.689888i\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (5.01 + 2.89i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.09 - 5.36i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.984 + 0.568i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.69 + 9.86i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 31.2iT - 289T^{2} \)
19 \( 1 + 32.9T + 361T^{2} \)
23 \( 1 + (29.1 + 16.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (22.1 - 12.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.5 + 20.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 8.80T + 1.36e3T^{2} \)
41 \( 1 + (61.2 + 35.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-29.8 - 51.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-52.1 + 30.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 19.7iT - 2.80e3T^{2} \)
59 \( 1 + (-72.2 - 41.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.3 + 26.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.29 + 9.16i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 3.63iT - 5.04e3T^{2} \)
73 \( 1 + 17.2T + 5.32e3T^{2} \)
79 \( 1 + (56.6 + 98.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-10.2 + 5.90i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + (37.1 + 64.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.29609970323633214294361109875, −10.20629978246633818513647399553, −8.888134885614731912292165546243, −8.357803526440064916469699524484, −7.42529075175679457090639122842, −6.05979840076282472089418929350, −4.93743436570500794673182601322, −3.95408064236994023297875798467, −2.39001823642534088626382045758, −0.36084998951477977683199397301, 1.81446650370707381432019965373, 3.83546310042165985967292192158, 4.17233561316921489330267223862, 6.02134398212150232020580324502, 6.96769976944202534899349531391, 7.959079433455517931441128700531, 8.627582776333330260441287657063, 10.15237121093501082310034249677, 10.85161412249160596669261465638, 11.54475874322707229029171364238

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