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Properties

Label 2-108-4.3-c2-0-1
Degree $2$
Conductor $108$
Sign $-0.780 - 0.624i$
Analytic cond. $2.94278$
Root an. cond. $1.71545$
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  + (−1.80 + 0.866i)2-s + (2.49 − 3.12i)4-s − 3.60·5-s + 6.24i·7-s + (−1.80 + 7.79i)8-s + (6.49 − 3.12i)10-s + 12.1i·11-s − 16·13-s + (−5.40 − 11.2i)14-s + (−3.49 − 15.6i)16-s − 14.4·17-s + 37.4i·19-s + (−9.01 + 11.2i)20-s + (−10.5 − 21.8i)22-s − 17.3i·23-s + ⋯
L(s)  = 1  + (−0.901 + 0.433i)2-s + (0.624 − 0.780i)4-s − 0.721·5-s + 0.892i·7-s + (−0.225 + 0.974i)8-s + (0.649 − 0.312i)10-s + 1.10i·11-s − 1.23·13-s + (−0.386 − 0.804i)14-s + (−0.218 − 0.975i)16-s − 0.848·17-s + 1.97i·19-s + (−0.450 + 0.562i)20-s + (−0.477 − 0.993i)22-s − 0.753i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.780 - 0.624i$
Analytic conductor: \(2.94278\)
Root analytic conductor: \(1.71545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :1),\ -0.780 - 0.624i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.162542 + 0.463082i\)
\(L(\frac12)\) \(\approx\) \(0.162542 + 0.463082i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.80 - 0.866i)T \)
3 \( 1 \)
good5 \( 1 + 3.60T + 25T^{2} \)
7 \( 1 - 6.24iT - 49T^{2} \)
11 \( 1 - 12.1iT - 121T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + 14.4T + 289T^{2} \)
19 \( 1 - 37.4iT - 361T^{2} \)
23 \( 1 + 17.3iT - 529T^{2} \)
29 \( 1 - 50.4T + 841T^{2} \)
31 \( 1 + 6.24iT - 961T^{2} \)
37 \( 1 - 26T + 1.36e3T^{2} \)
41 \( 1 - 7.21T + 1.68e3T^{2} \)
43 \( 1 + 12.4iT - 1.84e3T^{2} \)
47 \( 1 + 3.46iT - 2.20e3T^{2} \)
53 \( 1 + 68.5T + 2.80e3T^{2} \)
59 \( 1 - 76.2iT - 3.48e3T^{2} \)
61 \( 1 - 8T + 3.72e3T^{2} \)
67 \( 1 + 62.4iT - 4.48e3T^{2} \)
71 \( 1 + 62.3iT - 5.04e3T^{2} \)
73 \( 1 + 19T + 5.32e3T^{2} \)
79 \( 1 + 49.9iT - 6.24e3T^{2} \)
83 \( 1 - 116. iT - 6.88e3T^{2} \)
89 \( 1 + 79.3T + 7.92e3T^{2} \)
97 \( 1 - 119T + 9.40e3T^{2} \)
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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

−14.35113164242206932597925035129, −12.34739419689625194334014976651, −11.91175941427313497230622073118, −10.42366518625411486617198120282, −9.544815737424552403639507510323, −8.336093269801188959565054161199, −7.46882528415824450000985403325, −6.20076444368179309651030167239, −4.68562984155342302920608823570, −2.27374230627834457218730045950, 0.46232928552665835465722279946, 2.91032412142286833201772520686, 4.44549043215068436169425847112, 6.73432965609407110810964541662, 7.63806317630989470297968483700, 8.722936071687268655041231808498, 9.870983328941158326482560425221, 11.05914602867629248226365522907, 11.58521394104651800935152410205, 12.89932547068171856274387039118

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