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Properties

Label 2-18e2-27.4-c1-0-0
Degree $2$
Conductor $324$
Sign $-0.468 - 0.883i$
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  + (−0.583 + 3.31i)5-s + (−1.47 + 1.23i)7-s + (−0.352 − 1.99i)11-s + (−5.53 + 2.01i)13-s + (−1.69 + 2.93i)17-s + (−0.0802 − 0.138i)19-s + (3.29 + 2.76i)23-s + (−5.92 − 2.15i)25-s + (7.79 + 2.83i)29-s + (8.07 + 6.77i)31-s + (−3.23 − 5.59i)35-s + (2.17 − 3.77i)37-s + (−1.21 + 0.441i)41-s + (−1.25 − 7.11i)43-s + (−0.0743 + 0.0623i)47-s + ⋯
L(s)  = 1  + (−0.261 + 1.48i)5-s + (−0.556 + 0.467i)7-s + (−0.106 − 0.602i)11-s + (−1.53 + 0.558i)13-s + (−0.411 + 0.712i)17-s + (−0.0184 − 0.0318i)19-s + (0.686 + 0.576i)23-s + (−1.18 − 0.431i)25-s + (1.44 + 0.526i)29-s + (1.45 + 1.21i)31-s + (−0.546 − 0.946i)35-s + (0.358 − 0.620i)37-s + (−0.189 + 0.0689i)41-s + (−0.191 − 1.08i)43-s + (−0.0108 + 0.00909i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.468 - 0.883i$
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: \((2,\ 324,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465758 + 0.773944i\)
\(L(\frac12)\) \(\approx\) \(0.465758 + 0.773944i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.583 - 3.31i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.47 - 1.23i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.352 + 1.99i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (5.53 - 2.01i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.69 - 2.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0802 + 0.138i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.29 - 2.76i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-7.79 - 2.83i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-8.07 - 6.77i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-2.17 + 3.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.21 - 0.441i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.25 + 7.11i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.0743 - 0.0623i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (-0.422 + 2.39i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-4.41 + 3.70i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.89 + 1.41i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.09 + 5.36i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.12 - 3.68i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.80 - 1.02i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-8.95 - 3.26i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.821 - 1.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.484 - 2.74i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.87472352268292287233130301593, −10.91278971646435445382142497638, −10.20287215657883506782507163760, −9.228832936860385846660606287710, −8.039142315424790708108144361534, −6.89251435897548877907072926858, −6.39970991156028241760730947527, −4.92634901702302194465027349907, −3.36607877360622827767077813545, −2.51382983950418931216563433391, 0.63063924815701597821858785840, 2.69287961797166478885625853552, 4.52993734488169398349690233235, 4.91062155635789809796620258867, 6.48799501227482799739048240611, 7.60975719837470968430654685398, 8.438080981895372749388505070826, 9.627045414096748925544935668124, 10.02660998487945393441023954051, 11.55085233536513958245298199032

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