Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.639 - 0.768i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 3.76i)2-s + (−12.3 − 10.2i)4-s − 2.66·5-s + 88.8i·7-s + (−55.2 + 32.3i)8-s + (−3.61 + 10.0i)10-s + 72.9i·11-s − 173.·13-s + (334. + 120. i)14-s + (46.7 + 251. i)16-s + 423.·17-s − 153. i·19-s + (32.7 + 27.2i)20-s + (274. + 99.1i)22-s + 723. i·23-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (−0.768 − 0.639i)4-s − 0.106·5-s + 1.81i·7-s + (−0.862 + 0.505i)8-s + (−0.0361 + 0.100i)10-s + 0.602i·11-s − 1.02·13-s + (1.70 + 0.616i)14-s + (0.182 + 0.983i)16-s + 1.46·17-s − 0.424i·19-s + (0.0818 + 0.0680i)20-s + (0.567 + 0.204i)22-s + 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.639 - 0.768i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.639 - 0.768i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.09291 + 0.512630i\)
\(L(\frac12)\)  \(\approx\)  \(1.09291 + 0.512630i\)
\(L(3)\)   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 \( 1 + (-1.35 + 3.76i)T \)
3 \( 1 \)
good5 \( 1 + 2.66T + 625T^{2} \)
7 \( 1 - 88.8iT - 2.40e3T^{2} \)
11 \( 1 - 72.9iT - 1.46e4T^{2} \)
13 \( 1 + 173.T + 2.85e4T^{2} \)
17 \( 1 - 423.T + 8.35e4T^{2} \)
19 \( 1 + 153. iT - 1.30e5T^{2} \)
23 \( 1 - 723. iT - 2.79e5T^{2} \)
29 \( 1 + 366.T + 7.07e5T^{2} \)
31 \( 1 - 642. iT - 9.23e5T^{2} \)
37 \( 1 + 535.T + 1.87e6T^{2} \)
41 \( 1 - 2.14e3T + 2.82e6T^{2} \)
43 \( 1 - 615. iT - 3.41e6T^{2} \)
47 \( 1 - 330. iT - 4.87e6T^{2} \)
53 \( 1 + 4.62e3T + 7.89e6T^{2} \)
59 \( 1 + 3.46e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.64e3T + 1.38e7T^{2} \)
67 \( 1 - 6.67e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.44e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.73e3T + 2.83e7T^{2} \)
79 \( 1 - 2.08e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.29e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.70e3T + 6.27e7T^{2} \)
97 \( 1 - 6.97e3T + 8.85e7T^{2} \)
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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

−12.66592590979405497891950915182, −12.18242058288372671041672671627, −11.35821835831494814135137049638, −9.793755759609374243337495332569, −9.264366544421614796753692205916, −7.81492413100370455282339985916, −5.83018096171137779505308721868, −4.97442344850641632836181583272, −3.15925243047587622804431985635, −1.91189709319388451590974474890, 0.48189235055705601045811559503, 3.50933875305484591864886745151, 4.60372904589352252769149174766, 6.07583574441085805779492927778, 7.39585546558460978834180699360, 7.932535150382585784512635743762, 9.591991691805989128570425405019, 10.56221245096194048737990515893, 12.07488594128780005566942840781, 13.11263281362411988477523856922

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