Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.558 - 0.829i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 − 2.60i)2-s + (−0.629 − 5.15i)3-s + (−5.52 + 5.79i)4-s + (4.68 − 12.8i)5-s + (−12.7 + 7.38i)6-s + (−21.3 − 3.76i)7-s + (21.2 + 7.90i)8-s + (−26.2 + 6.49i)9-s + (−38.7 + 2.15i)10-s + (21.5 − 7.83i)11-s + (33.3 + 24.8i)12-s + (15.3 − 12.9i)13-s + (13.9 + 59.6i)14-s + (−69.3 − 16.0i)15-s + (−3.05 − 63.9i)16-s + (−99.6 + 57.5i)17-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (−0.121 − 0.992i)3-s + (−0.690 + 0.723i)4-s + (0.419 − 1.15i)5-s + (−0.864 + 0.502i)6-s + (−1.15 − 0.203i)7-s + (0.936 + 0.349i)8-s + (−0.970 + 0.240i)9-s + (−1.22 + 0.0680i)10-s + (0.590 − 0.214i)11-s + (0.802 + 0.597i)12-s + (0.328 − 0.275i)13-s + (0.266 + 1.13i)14-s + (−1.19 − 0.276i)15-s + (−0.0477 − 0.998i)16-s + (−1.42 + 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.558 - 0.829i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.558 - 0.829i)\)
\(L(2)\)  \(\approx\)  \(0.313109 + 0.587863i\)
\(L(\frac12)\)  \(\approx\)  \(0.313109 + 0.587863i\)
\(L(\frac{5}{2})\)   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 \( 1 + (1.11 + 2.60i)T \)
3 \( 1 + (0.629 + 5.15i)T \)
good5 \( 1 + (-4.68 + 12.8i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (21.3 + 3.76i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (-21.5 + 7.83i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-15.3 + 12.9i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (99.6 - 57.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-50.4 - 29.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-11.3 - 64.6i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-133. + 158. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (169. - 29.9i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (182. + 316. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (234. + 279. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (141. + 389. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-41.1 + 233. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 - 105. iT - 1.48e5T^{2} \)
59 \( 1 + (-537. - 195. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (-17.2 + 97.7i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (569. + 678. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-363. - 628. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (309. - 535. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-33.9 + 40.5i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (133. + 111. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (283. + 163. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-830. + 302. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−12.61048859722147339589159704782, −11.72937500597848725046081821355, −10.43110607543072452345754460158, −9.129540522717388307777648755915, −8.543349418802621765893616229451, −6.98874063671050517979468548711, −5.57311342853320199636174402022, −3.69889984741493897962099657309, −1.84943221314687454309032980629, −0.42066889847063238066354289393, 3.12145084068685391308225328079, 4.78276661861412214826965049247, 6.37259801350991550528504797250, 6.79044824615558628524224664730, 8.799871879541066745811368034989, 9.575290047598202081082978210776, 10.37698875966011133892979980095, 11.43182567966016848411606507159, 13.29289383355853939912658572667, 14.25386697896715625997593893083

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