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Properties

Label 2-18e2-36.31-c2-0-14
Degree $2$
Conductor $324$
Sign $0.973 - 0.230i$
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  + (0.461 − 1.94i)2-s + (−3.57 − 1.79i)4-s + (−2.18 + 3.78i)5-s + (−1.84 + 1.06i)7-s + (−5.14 + 6.12i)8-s + (6.36 + 6.00i)10-s + (10.9 − 6.30i)11-s + (−4.63 + 8.02i)13-s + (1.22 + 4.09i)14-s + (9.55 + 12.8i)16-s + 14.6·17-s + 34.5i·19-s + (14.6 − 9.61i)20-s + (−7.23 − 24.1i)22-s + (35.3 + 20.3i)23-s + ⋯
L(s)  = 1  + (0.230 − 0.973i)2-s + (−0.893 − 0.448i)4-s + (−0.437 + 0.757i)5-s + (−0.264 + 0.152i)7-s + (−0.642 + 0.766i)8-s + (0.636 + 0.600i)10-s + (0.993 − 0.573i)11-s + (−0.356 + 0.617i)13-s + (0.0874 + 0.292i)14-s + (0.597 + 0.802i)16-s + 0.861·17-s + 1.81i·19-s + (0.731 − 0.480i)20-s + (−0.329 − 1.09i)22-s + (1.53 + 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.973 - 0.230i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32562 + 0.154954i\)
\(L(\frac12)\) \(\approx\) \(1.32562 + 0.154954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.461 + 1.94i)T \)
3 \( 1 \)
good5 \( 1 + (2.18 - 3.78i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.84 - 1.06i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-10.9 + 6.30i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (4.63 - 8.02i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 14.6T + 289T^{2} \)
19 \( 1 - 34.5iT - 361T^{2} \)
23 \( 1 + (-35.3 - 20.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.52 + 16.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-0.860 - 0.496i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 66.4T + 1.36e3T^{2} \)
41 \( 1 + (12.9 - 22.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.4 + 21.0i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (30.1 - 17.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 12.2T + 2.80e3T^{2} \)
59 \( 1 + (-49.1 - 28.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.8 + 63.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-82.4 - 47.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 75.5iT - 5.04e3T^{2} \)
73 \( 1 + 56.7T + 5.32e3T^{2} \)
79 \( 1 + (64.5 - 37.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (56.7 - 32.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 150.T + 7.92e3T^{2} \)
97 \( 1 + (-56.4 - 97.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.53886928053903580519485995343, −10.65714602523010499214338192773, −9.717892056647876640305235712413, −8.917499287173819514977639263029, −7.67709928159777961773644837106, −6.44356555162943684238302550934, −5.33448710219370837991216117125, −3.80925445296704250371206333648, −3.19639043704215978629381936050, −1.47731365879822312960422143281, 0.65634448172124427371109602450, 3.26356869833129403931175475176, 4.56426255677271565687320329284, 5.23266527655571699388225899146, 6.72895131661110061547613036266, 7.29549949172533725625153419283, 8.598560723063363458819176925115, 9.094134716307998270923149240851, 10.21041496007305642412264983833, 11.62608129336537668992650791278

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