Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.849 - 0.528i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (14.0 − 24.3i)5-s + (75.7 + 131. i)7-s + (−138. − 240. i)11-s + (−291. + 505. i)13-s + 1.61e3·17-s + 1.36e3·19-s + (428. − 741. i)23-s + (1.16e3 + 2.02e3i)25-s + (4.26e3 + 7.39e3i)29-s + (−1.46e3 + 2.54e3i)31-s + 4.26e3·35-s + 4.03e3·37-s + (9.44e3 − 1.63e4i)41-s + (1.01e4 + 1.75e4i)43-s + (−147. − 256. i)47-s + ⋯
L(s)  = 1  + (0.251 − 0.435i)5-s + (0.583 + 1.01i)7-s + (−0.346 − 0.599i)11-s + (−0.479 + 0.829i)13-s + 1.35·17-s + 0.869·19-s + (0.168 − 0.292i)23-s + (0.373 + 0.646i)25-s + (0.942 + 1.63i)29-s + (−0.274 + 0.475i)31-s + 0.587·35-s + 0.484·37-s + (0.877 − 1.52i)41-s + (0.837 + 1.45i)43-s + (−0.00976 − 0.0169i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.849 - 0.528i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.849 - 0.528i)\)
\(L(3)\)  \(\approx\)  \(1.96380 + 0.560988i\)
\(L(\frac12)\)  \(\approx\)  \(1.96380 + 0.560988i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-14.0 + 24.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-75.7 - 131. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (138. + 240. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (291. - 505. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 1.61e3T + 1.41e6T^{2} \)
19 \( 1 - 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + (-428. + 741. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-4.26e3 - 7.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.46e3 - 2.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 4.03e3T + 6.93e7T^{2} \)
41 \( 1 + (-9.44e3 + 1.63e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.01e4 - 1.75e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (147. + 256. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 3.03e3T + 4.18e8T^{2} \)
59 \( 1 + (8.61e3 - 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.28e4 + 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.31e4 + 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.66e4T + 1.80e9T^{2} \)
73 \( 1 - 1.49e3T + 2.07e9T^{2} \)
79 \( 1 + (4.96e4 + 8.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.50e4 - 4.33e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + (-3.33e4 - 5.77e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.64546200465706980083806451776, −11.95341126333950707879420601070, −10.82548846865451823614251572185, −9.436246216233107564930119884093, −8.638445038967365811634698061169, −7.40258500858005118752096314553, −5.75940827220070840408626649197, −4.91581690720870324513421030316, −2.96648729107531819423617426535, −1.33267262102304111631616495452, 0.925270396552731578992928963350, 2.78668236542198874491203066340, 4.40736139761940738563755020042, 5.74064842366782150389261634015, 7.33123996484941992271333666951, 7.941183645644945763073659514181, 9.819471289170027544304161568489, 10.35746189741087140115471632150, 11.57012826321025902377894069209, 12.69853523410872143270176217529

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