Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.609 - 0.792i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 − 1.59i)2-s + (−0.204 − 2.99i)3-s + (−1.09 + 3.84i)4-s + (1.50 + 8.54i)5-s + (−4.53 + 3.93i)6-s + (−7.53 + 8.98i)7-s + (7.46 − 2.87i)8-s + (−8.91 + 1.22i)9-s + (11.8 − 12.6i)10-s + (2.27 + 0.400i)11-s + (11.7 + 2.50i)12-s + (−4.42 − 1.61i)13-s + (23.4 + 1.21i)14-s + (25.2 − 6.25i)15-s + (−13.5 − 8.45i)16-s + (6.77 + 11.7i)17-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.0681 − 0.997i)3-s + (−0.274 + 0.961i)4-s + (0.301 + 1.70i)5-s + (−0.755 + 0.655i)6-s + (−1.07 + 1.28i)7-s + (0.933 − 0.359i)8-s + (−0.990 + 0.136i)9-s + (1.18 − 1.26i)10-s + (0.206 + 0.0364i)11-s + (0.977 + 0.208i)12-s + (−0.340 − 0.123i)13-s + (1.67 + 0.0870i)14-s + (1.68 − 0.417i)15-s + (−0.848 − 0.528i)16-s + (0.398 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.609 - 0.792i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.609 - 0.792i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.593086 + 0.292092i\)
\(L(\frac12)\)  \(\approx\)  \(0.593086 + 0.292092i\)
\(L(2)\)   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 \( 1 + (1.20 + 1.59i)T \)
3 \( 1 + (0.204 + 2.99i)T \)
good5 \( 1 + (-1.50 - 8.54i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (7.53 - 8.98i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (-2.27 - 0.400i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (4.42 + 1.61i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-6.77 - 11.7i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (13.9 + 8.02i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.20 - 3.82i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-34.0 + 12.4i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-0.283 - 0.337i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-18.9 - 32.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-2.35 - 0.858i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-17.5 - 3.10i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (5.64 - 6.72i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 41.4T + 2.80e3T^{2} \)
59 \( 1 + (25.5 - 4.51i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-51.1 - 42.9i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (32.3 - 88.9i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-100. + 57.8i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-22.5 + 39.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-19.8 - 54.4i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-22.8 - 62.8i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-54.3 + 94.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-30.2 + 171. i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.29708963654529493317121518229, −12.41336985175731944769622233669, −11.56811327861037862601285873445, −10.49773128514088353549476491618, −9.484942115189624578535085679886, −8.213197079373895924518887549897, −6.88471549310296117267621066704, −6.11968311064147020267769499318, −3.10617942956069680141545044305, −2.35075362219818426939604611654, 0.58593347148815828340774936966, 4.15811266757269803460760843153, 5.14606655416277596144409219378, 6.46132613771650418889456648178, 8.040813507930541536832515553010, 9.205153754715341918898217294248, 9.727181127289526059596194821492, 10.69400000124055010048001245223, 12.41664759254866182755362083747, 13.55339014028637128285968609673

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