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Properties

Label 2-108-36.23-c5-0-7
Degree $2$
Conductor $108$
Sign $-0.531 - 0.847i$
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  + (−5.48 + 1.40i)2-s + (28.0 − 15.3i)4-s + (77.8 + 44.9i)5-s + (−124. + 71.6i)7-s + (−132. + 123. i)8-s + (−489. − 137. i)10-s + (278. + 481. i)11-s + (179. − 311. i)13-s + (579. − 566. i)14-s + (551. − 863. i)16-s − 1.07e3i·17-s + 348. i·19-s + (2.87e3 + 64.1i)20-s + (−2.20e3 − 2.25e3i)22-s + (−1.46e3 + 2.54e3i)23-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)2-s + (0.876 − 0.480i)4-s + (1.39 + 0.803i)5-s + (−0.956 + 0.552i)7-s + (−0.730 + 0.683i)8-s + (−1.54 − 0.433i)10-s + (0.693 + 1.20i)11-s + (0.295 − 0.511i)13-s + (0.789 − 0.772i)14-s + (0.538 − 0.842i)16-s − 0.903i·17-s + 0.221i·19-s + (1.60 + 0.0358i)20-s + (−0.969 − 0.991i)22-s + (−0.579 + 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.531 - 0.847i$
Analytic conductor: \(17.3214\)
Root analytic conductor: \(4.16190\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :5/2),\ -0.531 - 0.847i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.561719 + 1.01568i\)
\(L(\frac12)\) \(\approx\) \(0.561719 + 1.01568i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.48 - 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (-77.8 - 44.9i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (124. - 71.6i)T + (8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-278. - 481. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-179. + 311. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 1.07e3iT - 1.41e6T^{2} \)
19 \( 1 - 348. iT - 2.47e6T^{2} \)
23 \( 1 + (1.46e3 - 2.54e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (3.33e3 - 1.92e3i)T + (1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (2.34e3 + 1.35e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + 1.99e3T + 6.93e7T^{2} \)
41 \( 1 + (-1.34e4 - 7.75e3i)T + (5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (1.24e4 - 7.20e3i)T + (7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-1.23e3 - 2.13e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.33e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.98e4 - 3.44e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (211. + 366. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.40e4 - 8.09e3i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.90e3T + 1.80e9T^{2} \)
73 \( 1 + 1.10e4T + 2.07e9T^{2} \)
79 \( 1 + (-6.43e4 + 3.71e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (4.85e4 + 8.41e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 7.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.79e4 - 8.30e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.14229898907884952245902741328, −11.90025268689152044849147362319, −10.60188057665145714262751786097, −9.553718232763337723345281868538, −9.377870869531242839436788293619, −7.44590832871769979648959998464, −6.45807330485886701458470840505, −5.64430869697842445134739973217, −2.91925621112588827101907829953, −1.71109807213527851676387532027, 0.58754375194353222097981950573, 1.90745470664867267850874898977, 3.68121692405339250097202387973, 5.92602806389259442509766780477, 6.64131960353355876993205060363, 8.452170936620675874785928297790, 9.221631084689165572379481469223, 10.04702845329270974852019405772, 11.04738427139433754179079263528, 12.46302576474827104171848491414

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