Properties

Label 2-18e2-108.103-c2-0-27
Degree $2$
Conductor $324$
Sign $0.987 - 0.156i$
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  + (1.96 + 0.393i)2-s + (3.69 + 1.54i)4-s + (6.55 − 2.38i)5-s + (2.70 − 0.477i)7-s + (6.62 + 4.48i)8-s + (13.7 − 2.09i)10-s + (−2.46 + 6.77i)11-s + (−14.3 − 12.0i)13-s + (5.49 + 0.129i)14-s + (11.2 + 11.3i)16-s + (9.12 − 15.8i)17-s + (−22.9 + 13.2i)19-s + (27.8 + 1.31i)20-s + (−7.50 + 12.3i)22-s + (8.61 + 1.51i)23-s + ⋯
L(s)  = 1  + (0.980 + 0.196i)2-s + (0.922 + 0.385i)4-s + (1.31 − 0.477i)5-s + (0.386 − 0.0681i)7-s + (0.828 + 0.560i)8-s + (1.37 − 0.209i)10-s + (−0.224 + 0.616i)11-s + (−1.10 − 0.923i)13-s + (0.392 + 0.00926i)14-s + (0.702 + 0.712i)16-s + (0.536 − 0.929i)17-s + (−1.20 + 0.695i)19-s + (1.39 + 0.0658i)20-s + (−0.341 + 0.560i)22-s + (0.374 + 0.0660i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.987 - 0.156i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.67382 + 0.288785i\)
\(L(\frac12)\) \(\approx\) \(3.67382 + 0.288785i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.393i)T \)
3 \( 1 \)
good5 \( 1 + (-6.55 + 2.38i)T + (19.1 - 16.0i)T^{2} \)
7 \( 1 + (-2.70 + 0.477i)T + (46.0 - 16.7i)T^{2} \)
11 \( 1 + (2.46 - 6.77i)T + (-92.6 - 77.7i)T^{2} \)
13 \( 1 + (14.3 + 12.0i)T + (29.3 + 166. i)T^{2} \)
17 \( 1 + (-9.12 + 15.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (22.9 - 13.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.61 - 1.51i)T + (497. + 180. i)T^{2} \)
29 \( 1 + (6.25 - 5.24i)T + (146. - 828. i)T^{2} \)
31 \( 1 + (-40.7 - 7.18i)T + (903. + 328. i)T^{2} \)
37 \( 1 + (18.4 - 32.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (31.3 + 26.3i)T + (291. + 1.65e3i)T^{2} \)
43 \( 1 + (12.6 - 34.8i)T + (-1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (24.9 - 4.39i)T + (2.07e3 - 755. i)T^{2} \)
53 \( 1 - 90.9T + 2.80e3T^{2} \)
59 \( 1 + (24.5 + 67.3i)T + (-2.66e3 + 2.23e3i)T^{2} \)
61 \( 1 + (11.9 + 67.9i)T + (-3.49e3 + 1.27e3i)T^{2} \)
67 \( 1 + (38.7 - 46.1i)T + (-779. - 4.42e3i)T^{2} \)
71 \( 1 + (51.1 + 29.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-35.2 - 61.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (95.9 + 114. i)T + (-1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (46.7 + 55.6i)T + (-1.19e3 + 6.78e3i)T^{2} \)
89 \( 1 + (-7.52 - 13.0i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (101. + 37.1i)T + (7.20e3 + 6.04e3i)T^{2} \)
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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

−11.72653853779589142769135809303, −10.34766198787657448421657922977, −9.879411421139443201211853732867, −8.421203212630263824745088445733, −7.40241859414381956943028707359, −6.30584962198017185674736087918, −5.24479221297397910190207958945, −4.73018709013578786236718654515, −2.90774438331300460052763312355, −1.76309385441217033984685212765, 1.83327116961825349834982661563, 2.74628857512366061137906931982, 4.33096345395997634816813867687, 5.42683556302601409697548671524, 6.28844222159897642616161013633, 7.11378061622592450633978411267, 8.590736783186202036457237838749, 9.894858072067704041369729759622, 10.49324677278236605952504756968, 11.43379902531321826162150520257

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