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Properties

Label 2-18e2-27.4-c1-0-1
Degree $2$
Conductor $324$
Sign $0.828 + 0.559i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.103 − 0.584i)5-s + (2.18 − 1.83i)7-s + (−0.0708 − 0.402i)11-s + (−0.182 + 0.0664i)13-s + (3.66 − 6.34i)17-s + (2.06 + 3.57i)19-s + (−3.12 − 2.61i)23-s + (4.36 + 1.58i)25-s + (9.66 + 3.51i)29-s + (−4.78 − 4.01i)31-s + (−0.848 − 1.46i)35-s + (−2.88 + 4.99i)37-s + (−7.92 + 2.88i)41-s + (−1.03 − 5.89i)43-s + (−8.62 + 7.23i)47-s + ⋯
L(s)  = 1  + (0.0461 − 0.261i)5-s + (0.826 − 0.693i)7-s + (−0.0213 − 0.121i)11-s + (−0.0506 + 0.0184i)13-s + (0.888 − 1.53i)17-s + (0.473 + 0.820i)19-s + (−0.650 − 0.546i)23-s + (0.873 + 0.317i)25-s + (1.79 + 0.653i)29-s + (−0.858 − 0.720i)31-s + (−0.143 − 0.248i)35-s + (−0.474 + 0.821i)37-s + (−1.23 + 0.450i)41-s + (−0.158 − 0.899i)43-s + (−1.25 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.828 + 0.559i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ 0.828 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37199 - 0.419687i\)
\(L(\frac12)\) \(\approx\) \(1.37199 - 0.419687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.103 + 0.584i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.18 + 1.83i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.0708 + 0.402i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.182 - 0.0664i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.66 + 6.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.06 - 3.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 2.61i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-9.66 - 3.51i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.78 + 4.01i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (2.88 - 4.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.92 - 2.88i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.03 + 5.89i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.62 - 7.23i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 3.42T + 53T^{2} \)
59 \( 1 + (0.813 - 4.61i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.34 + 2.80i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (11.4 - 4.15i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.09 - 7.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.96 - 3.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.47 + 3.08i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-7.91 - 2.88i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (2.38 + 4.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.34 - 7.62i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.63298533224465691257919112283, −10.53532286148079818346976965888, −9.800200589001299244246347966886, −8.593158394502377034864823536333, −7.75067042415610654877071405973, −6.82516461881603048331890503028, −5.38104511501821582017872474449, −4.55462577439945083670558628589, −3.11097665369455811935498637790, −1.24498817761123715600079243800, 1.78361214897458252848519542946, 3.27980976170813475748140626552, 4.76117582000278561937734936036, 5.73076226891736116480530277473, 6.86610407829406518577965090609, 8.076685800676531826049747668513, 8.707198584877880150484563644733, 9.975805086252041071376921476601, 10.74421617735924104519408263875, 11.79850897775634696581030731532

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