Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.751 - 0.659i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.78 + 5.36i)2-s + (−25.6 − 19.1i)4-s + (52.1 − 30.1i)5-s + (−185. − 106. i)7-s + (148. − 103. i)8-s + (68.4 + 333. i)10-s + (−185. + 321. i)11-s + (402. + 697. i)13-s + (905. − 803. i)14-s + (287. + 982. i)16-s + 34.0i·17-s + 1.17e3i·19-s + (−1.91e3 − 229. i)20-s + (−1.39e3 − 1.57e3i)22-s + (1.98e3 + 3.43e3i)23-s + ⋯
L(s)  = 1  + (−0.316 + 0.948i)2-s + (−0.800 − 0.599i)4-s + (0.933 − 0.538i)5-s + (−1.42 − 0.825i)7-s + (0.821 − 0.569i)8-s + (0.216 + 1.05i)10-s + (−0.462 + 0.800i)11-s + (0.660 + 1.14i)13-s + (1.23 − 1.09i)14-s + (0.280 + 0.959i)16-s + 0.0285i·17-s + 0.745i·19-s + (−1.07 − 0.128i)20-s + (−0.613 − 0.691i)22-s + (0.780 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.751 - 0.659i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.751 - 0.659i)\)
\(L(3)\)  \(\approx\)  \(0.335134 + 0.890191i\)
\(L(\frac12)\)  \(\approx\)  \(0.335134 + 0.890191i\)
\(L(\frac{7}{2})\)   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 \( 1 + (1.78 - 5.36i)T \)
3 \( 1 \)
good5 \( 1 + (-52.1 + 30.1i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (185. + 106. i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (185. - 321. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-402. - 697. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 34.0iT - 1.41e6T^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + (-1.98e3 - 3.43e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-382. - 221. i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.38e3 - 797. i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.10e4T + 6.93e7T^{2} \)
41 \( 1 + (-7.96e3 + 4.60e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.06e3 - 612. i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.22e3 + 2.11e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 2.02e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.70e3 + 2.95e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.99e4 - 3.45e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-6.08e3 + 3.51e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.56e4T + 1.80e9T^{2} \)
73 \( 1 - 6.54e4T + 2.07e9T^{2} \)
79 \( 1 + (4.11e4 + 2.37e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-6.60e3 + 1.14e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 8.56e4iT - 5.58e9T^{2} \)
97 \( 1 + (2.79e4 - 4.84e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.45816092806771037759367528608, −12.57052222051932484570874637665, −10.56534059123260393644562129015, −9.630720776463366690107280743258, −9.080217868111991597721047998637, −7.40706140193804714401892453594, −6.50936625687570393647406763400, −5.41019310881327346149273146388, −3.90325415787114391931846552387, −1.40997128197197458379556556855, 0.42018229341237503159097767783, 2.53736998415292540090140593433, 3.24549213836145894789333103326, 5.44379330947294032093867456505, 6.53743165854700060806647660739, 8.396875559960581626151722419821, 9.332312020689725570629790706129, 10.28990706558600287251113845213, 11.02413528277076578847930396564, 12.56372163603523104540716655151

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