Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.177 + 0.984i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (2.73 + 2.92i)2-s + (−1.08 + 15.9i)4-s + (−19.5 − 33.8i)5-s + (−10.5 − 6.10i)7-s + (−49.6 + 40.4i)8-s + (45.5 − 149. i)10-s + (−96.1 − 55.5i)11-s + (−104. − 180. i)13-s + (−11.0 − 47.5i)14-s + (−253. − 34.7i)16-s − 93.3·17-s + 26.8i·19-s + (561. − 275. i)20-s + (−100. − 432. i)22-s + (757. − 437. i)23-s + ⋯
L(s)  = 1  + (0.682 + 0.730i)2-s + (−0.0680 + 0.997i)4-s + (−0.781 − 1.35i)5-s + (−0.215 − 0.124i)7-s + (−0.775 + 0.631i)8-s + (0.455 − 1.49i)10-s + (−0.794 − 0.458i)11-s + (−0.618 − 1.07i)13-s + (−0.0562 − 0.242i)14-s + (−0.990 − 0.135i)16-s − 0.323·17-s + 0.0744i·19-s + (1.40 − 0.687i)20-s + (−0.207 − 0.893i)22-s + (1.43 − 0.826i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.177 + 0.984i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.177 + 0.984i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.533653 - 0.638772i\)
\(L(\frac12)\)  \(\approx\)  \(0.533653 - 0.638772i\)
\(L(3)\)   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 \( 1 + (-2.73 - 2.92i)T \)
3 \( 1 \)
good5 \( 1 + (19.5 + 33.8i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (10.5 + 6.10i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (96.1 + 55.5i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (104. + 180. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 93.3T + 8.35e4T^{2} \)
19 \( 1 - 26.8iT - 1.30e5T^{2} \)
23 \( 1 + (-757. + 437. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (650. - 1.12e3i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-593. + 342. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.76e3T + 1.87e6T^{2} \)
41 \( 1 + (-39.0 - 67.6i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.40e3 + 811. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-1.99e3 - 1.15e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 1.31e3T + 7.89e6T^{2} \)
59 \( 1 + (4.81e3 - 2.78e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.09e3 - 1.88e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (213. - 123. i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 4.60e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.56e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.48e3 - 2.59e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-1.62e3 - 936. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.16e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.86e3 + 4.97e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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

−12.77930752408363651411988118395, −12.16632885225982800723460645091, −10.76429783478544126079978793342, −8.963805136133654056058834250669, −8.190049701227254377350929102333, −7.16829874516147494968229214838, −5.45360272959469313184432294391, −4.69201490262310496336733379630, −3.19023194676080291335178733449, −0.28169483272763404593417492724, 2.33304381480465380532885384453, 3.49345262076048686694947453158, 4.86246837723885878417865651971, 6.52506931168487381071292922793, 7.44757221286683241915498021598, 9.343698595606786218444110489455, 10.43241162128717605563044546562, 11.29262392606600431020735383331, 12.04669421110214470744974903119, 13.26755778698928787762341898063

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