Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.970 - 0.241i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (4.45 + 3.48i)2-s + (7.72 + 31.0i)4-s − 40.4i·5-s − 148. i·7-s + (−73.7 + 165. i)8-s + (141. − 180. i)10-s + 608.·11-s + 883.·13-s + (517. − 661. i)14-s + (−904. + 479. i)16-s − 1.13e3i·17-s + 1.11e3i·19-s + (1.25e3 − 312. i)20-s + (2.71e3 + 2.11e3i)22-s − 1.34e3·23-s + ⋯
L(s)  = 1  + (0.787 + 0.615i)2-s + (0.241 + 0.970i)4-s − 0.724i·5-s − 1.14i·7-s + (−0.407 + 0.913i)8-s + (0.446 − 0.570i)10-s + 1.51·11-s + 1.45·13-s + (0.705 − 0.902i)14-s + (−0.883 + 0.468i)16-s − 0.951i·17-s + 0.706i·19-s + (0.702 − 0.174i)20-s + (1.19 + 0.933i)22-s − 0.530·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.970 - 0.241i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.970 - 0.241i)\)
\(L(3)\)  \(\approx\)  \(3.15955 + 0.386951i\)
\(L(\frac12)\)  \(\approx\)  \(3.15955 + 0.386951i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-4.45 - 3.48i)T \)
3 \( 1 \)
good5 \( 1 + 40.4iT - 3.12e3T^{2} \)
7 \( 1 + 148. iT - 1.68e4T^{2} \)
11 \( 1 - 608.T + 1.61e5T^{2} \)
13 \( 1 - 883.T + 3.71e5T^{2} \)
17 \( 1 + 1.13e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.11e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.34e3T + 6.43e6T^{2} \)
29 \( 1 - 1.85e3iT - 2.05e7T^{2} \)
31 \( 1 + 606. iT - 2.86e7T^{2} \)
37 \( 1 - 1.36e4T + 6.93e7T^{2} \)
41 \( 1 + 2.12e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.24e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.62e4T + 2.29e8T^{2} \)
53 \( 1 + 1.34e3iT - 4.18e8T^{2} \)
59 \( 1 + 3.12e4T + 7.14e8T^{2} \)
61 \( 1 - 6.50e3T + 8.44e8T^{2} \)
67 \( 1 - 2.35e3iT - 1.35e9T^{2} \)
71 \( 1 - 2.69e4T + 1.80e9T^{2} \)
73 \( 1 + 5.23e4T + 2.07e9T^{2} \)
79 \( 1 - 3.57e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.29e4T + 3.93e9T^{2} \)
89 \( 1 + 5.45e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.42e5T + 8.58e9T^{2} \)
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

−13.03938572785576755603173759237, −11.95478295400473818894673823741, −10.99533415696947377857433577903, −9.319476480055264050077980826375, −8.257407092126953737869619561676, −7.02326014244676411406852327190, −5.99294454313270033859191899688, −4.45601626340765157938642649826, −3.63253587966488037086468834657, −1.15818913962509286636950076318, 1.49213774125702166260525825581, 2.98815528118352734369503820628, 4.20036304031634893693805742987, 5.97000992485310458369624072825, 6.52617192826649204047253926057, 8.601800438640013913750542779805, 9.652040175882788332083707737084, 11.02593137361109585024961861442, 11.59315601178496258570143447350, 12.65911545226025071065500913937

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