Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.293 + 0.956i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

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

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.293 + 0.956i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.293 + 0.956i)\)
\(L(2)\)  \(\approx\)  \(0.884762 - 0.654117i\)
\(L(\frac12)\)  \(\approx\)  \(0.884762 - 0.654117i\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

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

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