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Properties

Label 2-108-36.11-c5-0-3
Degree $2$
Conductor $108$
Sign $-0.997 + 0.0720i$
Analytic cond. $17.3214$
Root an. cond. $4.16190$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 + 4.22i)2-s + (−3.74 − 31.7i)4-s + (−12.8 + 7.42i)5-s + (15.2 + 8.83i)7-s + (148. + 103. i)8-s + (16.9 − 82.2i)10-s + (143. − 248. i)11-s + (42.2 + 73.1i)13-s + (−94.8 + 31.4i)14-s + (−995. + 238. i)16-s + 1.60e3i·17-s + 1.66e3i·19-s + (284. + 380. i)20-s + (510. + 1.53e3i)22-s + (−2.32e3 − 4.02e3i)23-s + ⋯
L(s)  = 1  + (−0.664 + 0.747i)2-s + (−0.117 − 0.993i)4-s + (−0.230 + 0.132i)5-s + (0.118 + 0.0681i)7-s + (0.820 + 0.572i)8-s + (0.0535 − 0.260i)10-s + (0.356 − 0.618i)11-s + (0.0693 + 0.120i)13-s + (−0.129 + 0.0429i)14-s + (−0.972 + 0.232i)16-s + 1.35i·17-s + 1.06i·19-s + (0.158 + 0.212i)20-s + (0.224 + 0.677i)22-s + (−0.916 − 1.58i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0720i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.997 + 0.0720i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ -0.997 + 0.0720i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0152527 - 0.422666i\)
\(L(\frac12)\) \(\approx\) \(0.0152527 - 0.422666i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.75 - 4.22i)T \)
3 \( 1 \)
good5 \( 1 + (12.8 - 7.42i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-15.2 - 8.83i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-143. + 248. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-42.2 - 73.1i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.60e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.66e3iT - 2.47e6T^{2} \)
23 \( 1 + (2.32e3 + 4.02e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.41e3 + 1.39e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (2.78e3 - 1.60e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 4.62e3T + 6.93e7T^{2} \)
41 \( 1 + (9.81e3 - 5.66e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (1.23e4 + 7.15e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.69e3 - 2.93e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.10e4iT - 4.18e8T^{2} \)
59 \( 1 + (-1.09e4 - 1.88e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.78e4 - 3.09e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.80e4 + 1.04e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.22e4T + 1.80e9T^{2} \)
73 \( 1 + 7.29e4T + 2.07e9T^{2} \)
79 \( 1 + (-7.60e4 - 4.39e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (1.22e4 - 2.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.19e5iT - 5.58e9T^{2} \)
97 \( 1 + (-3.95e4 + 6.85e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−13.56624388648512117716166167684, −12.15299971684637493921582256137, −10.90492261701397400586099032437, −10.04569952303488989642758159254, −8.698634461340590389206246330551, −7.984964131748723899424912549368, −6.60591801626703389676393990839, −5.64512858577822627961378439723, −3.94834728862762231653803296793, −1.64413225779144543966506216347, 0.19903525563218961719765116226, 1.88486204321615420677226408105, 3.49033900805162453352156096974, 4.89526648428142910306597789707, 6.95337801793560392393797305437, 7.930016980530528904045578612872, 9.208803135153396240672716442122, 9.949299684027866723867990638666, 11.33262175929106382790271801757, 11.89016007437557528892206235675

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