Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.532 - 0.846i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.485 − 3.97i)2-s + (−15.5 − 3.85i)4-s + (−1.01 + 1.75i)5-s + (−20.0 + 11.5i)7-s + (−22.8 + 59.7i)8-s + (6.48 + 4.88i)10-s + (4.32 − 2.49i)11-s + (−137. + 238. i)13-s + (36.1 + 85.1i)14-s + (226. + 119. i)16-s + 266.·17-s + 367. i·19-s + (22.5 − 23.3i)20-s + (−7.82 − 18.4i)22-s + (−544. − 314. i)23-s + ⋯
L(s)  = 1  + (0.121 − 0.992i)2-s + (−0.970 − 0.241i)4-s + (−0.0406 + 0.0703i)5-s + (−0.408 + 0.236i)7-s + (−0.357 + 0.934i)8-s + (0.0648 + 0.0488i)10-s + (0.0357 − 0.0206i)11-s + (−0.815 + 1.41i)13-s + (0.184 + 0.434i)14-s + (0.883 + 0.468i)16-s + 0.920·17-s + 1.01i·19-s + (0.0563 − 0.0584i)20-s + (−0.0161 − 0.0380i)22-s + (−1.02 − 0.594i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.532 - 0.846i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.532 - 0.846i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.701796 + 0.387815i\)
\(L(\frac12)\)  \(\approx\)  \(0.701796 + 0.387815i\)
\(L(3)\)   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 \( 1 + (-0.485 + 3.97i)T \)
3 \( 1 \)
good5 \( 1 + (1.01 - 1.75i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (20.0 - 11.5i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-4.32 + 2.49i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (137. - 238. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 266.T + 8.35e4T^{2} \)
19 \( 1 - 367. iT - 1.30e5T^{2} \)
23 \( 1 + (544. + 314. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-319. - 553. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (1.19e3 + 687. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + (593. - 1.02e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.43e3 - 825. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-307. + 177. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 5.29e3T + 7.89e6T^{2} \)
59 \( 1 + (5.22e3 + 3.01e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (833. + 1.44e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.90e3 - 1.10e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 + (4.44e3 - 2.56e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-6.91e3 + 3.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 8.86e3T + 6.27e7T^{2} \)
97 \( 1 + (3.40e3 + 5.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−12.83573883207989855439389874743, −12.14023046201381148119972454756, −11.18567056727351873736566040604, −9.922480421105273111822821281275, −9.256181041877684022946194030502, −7.83592687518272534995990296740, −6.15903999274520581729756499144, −4.69719858466702961881343657457, −3.33769499355989267310674645402, −1.76807053971043331931485173629, 0.34437484738428526533796930870, 3.25331747735367252077632760722, 4.83677664619365883427878609828, 5.97405799765847474913230872841, 7.28291224098940263052753653842, 8.150496841201232401650599673334, 9.500159554265428375048274621443, 10.38526881554314037590647470931, 12.15707089639463445866905968021, 12.96172723750360602340442169023

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