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Properties

Label 2-6e3-72.5-c2-0-15
Degree $2$
Conductor $216$
Sign $-0.619 + 0.784i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
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 − 0.355i)2-s + (3.74 + 1.39i)4-s + (4.28 − 7.41i)5-s + (−3.75 − 6.50i)7-s + (−6.87 − 4.08i)8-s + (−11.0 + 13.0i)10-s + (4.74 + 8.22i)11-s + (−9.54 − 5.51i)13-s + (5.08 + 14.1i)14-s + (12.0 + 10.4i)16-s − 11.3i·17-s + 18.3i·19-s + (26.4 − 21.7i)20-s + (−6.41 − 17.8i)22-s + (−22.8 − 13.1i)23-s + ⋯
L(s)  = 1  + (−0.984 − 0.177i)2-s + (0.936 + 0.349i)4-s + (0.856 − 1.48i)5-s + (−0.536 − 0.929i)7-s + (−0.859 − 0.510i)8-s + (−1.10 + 1.30i)10-s + (0.431 + 0.747i)11-s + (−0.734 − 0.424i)13-s + (0.362 + 1.01i)14-s + (0.755 + 0.655i)16-s − 0.667i·17-s + 0.967i·19-s + (1.32 − 1.08i)20-s + (−0.291 − 0.812i)22-s + (−0.993 − 0.573i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.619 + 0.784i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.619 + 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.385274 - 0.795364i\)
\(L(\frac12)\) \(\approx\) \(0.385274 - 0.795364i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.355i)T \)
3 \( 1 \)
good5 \( 1 + (-4.28 + 7.41i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.75 + 6.50i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-4.74 - 8.22i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (9.54 + 5.51i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 11.3iT - 289T^{2} \)
19 \( 1 - 18.3iT - 361T^{2} \)
23 \( 1 + (22.8 + 13.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (3.48 + 6.03i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.42 + 11.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 5.89iT - 1.36e3T^{2} \)
41 \( 1 + (32.7 + 18.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-21.1 + 12.2i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-15.8 + 9.17i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 58.4T + 2.80e3T^{2} \)
59 \( 1 + (-13.1 + 22.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-56.0 + 32.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-28.4 - 16.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 84.5iT - 5.04e3T^{2} \)
73 \( 1 + 94.1T + 5.32e3T^{2} \)
79 \( 1 + (12.3 + 21.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-57.7 - 100. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.1 - 163. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.89989301723612990714005719889, −10.15974225372690139426077136092, −9.929329040062774185044513719115, −8.966231254257835121308984178425, −7.908482861243372214423491246176, −6.83417520162476580977220596238, −5.55910755572988727604896547499, −4.09114208966714530505922135542, −2.07386844520541052338092116516, −0.64449063635645273276994758104, 2.12695648373792257279697273035, 3.11146614787537981591555999123, 5.74437662790565378739282686033, 6.39925025129313749034542471376, 7.26790233242990077490196655500, 8.698895716882363072867691255652, 9.561015762480398474615941249144, 10.29534209034453437592440496427, 11.24481674407217835586021994928, 12.06869188918169000285280059389

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