Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.925 + 0.378i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.85 + 0.741i)2-s + (−2.48 + 1.67i)3-s + (2.90 − 2.75i)4-s + (1.04 + 5.95i)5-s + (3.37 − 4.96i)6-s + (−1.18 + 1.41i)7-s + (−3.34 + 7.26i)8-s + (3.36 − 8.34i)9-s + (−6.36 − 10.2i)10-s + (−14.9 − 2.63i)11-s + (−2.58 + 11.7i)12-s + (−6.81 − 2.48i)13-s + (1.15 − 3.50i)14-s + (−12.5 − 13.0i)15-s + (0.820 − 15.9i)16-s + (2.22 + 3.85i)17-s + ⋯
L(s)  = 1  + (−0.928 + 0.370i)2-s + (−0.828 + 0.559i)3-s + (0.725 − 0.688i)4-s + (0.209 + 1.19i)5-s + (0.562 − 0.826i)6-s + (−0.169 + 0.201i)7-s + (−0.417 + 0.908i)8-s + (0.374 − 0.927i)9-s + (−0.636 − 1.02i)10-s + (−1.36 − 0.239i)11-s + (−0.215 + 0.976i)12-s + (−0.524 − 0.190i)13-s + (0.0824 − 0.250i)14-s + (−0.839 − 0.869i)15-s + (0.0512 − 0.998i)16-s + (0.130 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.925 + 0.378i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.925 + 0.378i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0497958 - 0.253551i\)
\(L(\frac12)\)  \(\approx\)  \(0.0497958 - 0.253551i\)
\(L(2)\)   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 \( 1 + (1.85 - 0.741i)T \)
3 \( 1 + (2.48 - 1.67i)T \)
good5 \( 1 + (-1.04 - 5.95i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (1.18 - 1.41i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (14.9 + 2.63i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (6.81 + 2.48i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-2.22 - 3.85i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (30.1 + 17.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.19 - 9.77i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (26.3 - 9.59i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-34.0 - 40.5i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (-1.20 - 2.08i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (44.2 + 16.1i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-15.3 - 2.71i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-13.3 + 15.8i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 77.1T + 2.80e3T^{2} \)
59 \( 1 + (-69.2 + 12.2i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-25.1 - 21.0i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (25.3 - 69.6i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (81.7 - 47.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (42.1 - 72.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-0.961 - 2.64i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-19.3 - 53.0i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-55.7 + 96.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (13.0 - 73.9i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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

−14.52761517595628642045745037108, −12.87083811488190798356013932925, −11.42941392210786554546593957111, −10.55941036521967862107656904237, −10.16801357008633032921781732129, −8.756960172362105111490234165727, −7.24604253542275112685683104785, −6.33866021430089037830500501221, −5.17783003468857675170305236643, −2.76068737908701733278953715899, 0.26192830753461030908701017946, 2.05833438632981511518122591590, 4.69113620833712236856052700475, 6.13133128304892881160742076906, 7.55433316746914349159333399414, 8.415405807476753141518175985165, 9.824008948503641440690483852686, 10.65463127108044890450802548388, 11.87018341306556292224805425003, 12.77325647438005639735161289190

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