Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.0267 - 0.999i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.664 + 2.74i)2-s + (−7.11 − 3.65i)4-s + (14.2 + 8.25i)5-s + (19.2 − 11.1i)7-s + (14.7 − 17.1i)8-s + (−32.1 + 33.8i)10-s + (6.37 + 11.0i)11-s + (−11.1 + 19.3i)13-s + (17.7 + 60.3i)14-s + (37.3 + 51.9i)16-s + 117. i·17-s − 27.7i·19-s + (−71.5 − 110. i)20-s + (−34.6 + 10.1i)22-s + (17.5 − 30.4i)23-s + ⋯
L(s)  = 1  + (−0.234 + 0.972i)2-s + (−0.889 − 0.456i)4-s + (1.27 + 0.737i)5-s + (1.04 − 0.600i)7-s + (0.652 − 0.757i)8-s + (−1.01 + 1.06i)10-s + (0.174 + 0.302i)11-s + (−0.238 + 0.413i)13-s + (0.339 + 1.15i)14-s + (0.583 + 0.812i)16-s + 1.67i·17-s − 0.334i·19-s + (−0.800 − 1.24i)20-s + (−0.335 + 0.0988i)22-s + (0.159 − 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0267 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.0267 - 0.999i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.0267 - 0.999i)\)
\(L(2)\)  \(\approx\)  \(1.22648 + 1.19405i\)
\(L(\frac12)\)  \(\approx\)  \(1.22648 + 1.19405i\)
\(L(\frac{5}{2})\)   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 \( 1 + (0.664 - 2.74i)T \)
3 \( 1 \)
good5 \( 1 + (-14.2 - 8.25i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-19.2 + 11.1i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-6.37 - 11.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (11.1 - 19.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 117. iT - 4.91e3T^{2} \)
19 \( 1 + 27.7iT - 6.85e3T^{2} \)
23 \( 1 + (-17.5 + 30.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (1.01 - 0.584i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (119. + 68.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 + (13.2 + 7.65i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-361. + 208. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (116. + 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 180. iT - 1.48e5T^{2} \)
59 \( 1 + (313. - 543. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (382. + 661. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (113. + 65.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 22.6T + 3.57e5T^{2} \)
73 \( 1 - 387.T + 3.89e5T^{2} \)
79 \( 1 + (486. - 280. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (342. + 592. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 278. iT - 7.04e5T^{2} \)
97 \( 1 + (264. + 458. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.87386124105517342963900899960, −12.84843315766551462259341212396, −10.96021313602547441827481959016, −10.19805858180164658186292733947, −9.112403413814073445840614955101, −7.82445459380083098410569824267, −6.72084448942522084415863973255, −5.71854327891820513412418388038, −4.31302868995231844262584449880, −1.73697533590655057560116691915, 1.25741966306243671265690169112, 2.59684262565933651842884293654, 4.77285444104095791947096409121, 5.60939029169106377891142846389, 7.82481001933996095899107423110, 9.037537310888077187317629253646, 9.595842563155207537499299640056, 10.92234871975839329456265504867, 11.88247927324929178777994959231, 12.86163934812073691112268959971

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