# Properties

 Label 2-108-12.11-c3-0-13 Degree $2$ Conductor $108$ Sign $0.293 + 0.956i$ Analytic cond. $6.37220$ Root an. cond. $2.52432$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−2.27 − 1.68i)2-s + (2.34 + 7.64i)4-s − 5.83i·5-s + 8.83i·7-s + (7.52 − 21.3i)8-s + (−9.81 + 13.2i)10-s + 23.6·11-s + 54.6·13-s + (14.8 − 20.0i)14-s + (−52.9 + 35.8i)16-s − 117. i·17-s − 109. i·19-s + (44.6 − 13.6i)20-s + (−53.7 − 39.7i)22-s − 33.5·23-s + ⋯
 L(s)  = 1 + (−0.804 − 0.594i)2-s + (0.293 + 0.956i)4-s − 0.522i·5-s + 0.476i·7-s + (0.332 − 0.943i)8-s + (−0.310 + 0.419i)10-s + 0.647·11-s + 1.16·13-s + (0.283 − 0.383i)14-s + (−0.828 + 0.560i)16-s − 1.67i·17-s − 1.32i·19-s + (0.499 − 0.153i)20-s + (−0.520 − 0.384i)22-s − 0.304·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$108$$    =    $$2^{2} \cdot 3^{3}$$ Sign: $0.293 + 0.956i$ Analytic conductor: $$6.37220$$ Root analytic conductor: $$2.52432$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{108} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 108,\ (\ :3/2),\ 0.293 + 0.956i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.884762 - 0.654117i$$ $$L(\frac12)$$ $$\approx$$ $$0.884762 - 0.654117i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (2.27 + 1.68i)T$$
3 $$1$$
good5 $$1 + 5.83iT - 125T^{2}$$
7 $$1 - 8.83iT - 343T^{2}$$
11 $$1 - 23.6T + 1.33e3T^{2}$$
13 $$1 - 54.6T + 2.19e3T^{2}$$
17 $$1 + 117. iT - 4.91e3T^{2}$$
19 $$1 + 109. iT - 6.85e3T^{2}$$
23 $$1 + 33.5T + 1.21e4T^{2}$$
29 $$1 - 40.0iT - 2.43e4T^{2}$$
31 $$1 + 292. iT - 2.97e4T^{2}$$
37 $$1 - 283.T + 5.06e4T^{2}$$
41 $$1 - 367. iT - 6.89e4T^{2}$$
43 $$1 - 323. iT - 7.95e4T^{2}$$
47 $$1 - 66.2T + 1.03e5T^{2}$$
53 $$1 - 158. iT - 1.48e5T^{2}$$
59 $$1 + 848.T + 2.05e5T^{2}$$
61 $$1 + 348.T + 2.26e5T^{2}$$
67 $$1 + 194. iT - 3.00e5T^{2}$$
71 $$1 - 939.T + 3.57e5T^{2}$$
73 $$1 + 473.T + 3.89e5T^{2}$$
79 $$1 - 273. iT - 4.93e5T^{2}$$
83 $$1 - 338.T + 5.71e5T^{2}$$
89 $$1 + 739. iT - 7.04e5T^{2}$$
97 $$1 + 448.T + 9.12e5T^{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

−12.85345578334030400480052399516, −11.66424828608679418695471997670, −11.11134172480867025143975270423, −9.464516143864735537552768677011, −9.009274591021853781393164638661, −7.76425061994466178873698401722, −6.37798557578949725817566251373, −4.51604350393361402217808887671, −2.80286244317110914645129734090, −0.926458814277290342799473864912, 1.45882833618538772919744254467, 3.84583490796172822610150843420, 5.87510152103000944163804955359, 6.72753447090336494927462392907, 8.011531022393317762725357734680, 8.938009382994271515568611144527, 10.39668495115623900422042749896, 10.80416408607669562963564123381, 12.28697155864954245125356024701, 13.81757380807405142927817818370