Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.850 - 0.526i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.72 − 0.773i)2-s + (6.80 − 4.21i)4-s + 20.8i·5-s + 13.9i·7-s + (15.2 − 16.7i)8-s + (16.1 + 56.7i)10-s + 34.5·11-s − 31.3·13-s + (10.7 + 37.8i)14-s + (28.5 − 57.2i)16-s − 34.4i·17-s − 120. i·19-s + (87.7 + 141. i)20-s + (93.8 − 26.7i)22-s + 137.·23-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)2-s + (0.850 − 0.526i)4-s + 1.86i·5-s + 0.750i·7-s + (0.673 − 0.738i)8-s + (0.510 + 1.79i)10-s + 0.945·11-s − 0.668·13-s + (0.205 + 0.722i)14-s + (0.445 − 0.895i)16-s − 0.491i·17-s − 1.45i·19-s + (0.981 + 1.58i)20-s + (0.909 − 0.258i)22-s + 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.850 - 0.526i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.850 - 0.526i)\)
\(L(2)\)  \(\approx\)  \(2.70750 + 0.770263i\)
\(L(\frac12)\)  \(\approx\)  \(2.70750 + 0.770263i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-2.72 + 0.773i)T \)
3 \( 1 \)
good5 \( 1 - 20.8iT - 125T^{2} \)
7 \( 1 - 13.9iT - 343T^{2} \)
11 \( 1 - 34.5T + 1.33e3T^{2} \)
13 \( 1 + 31.3T + 2.19e3T^{2} \)
17 \( 1 + 34.4iT - 4.91e3T^{2} \)
19 \( 1 + 120. iT - 6.85e3T^{2} \)
23 \( 1 - 137.T + 1.21e4T^{2} \)
29 \( 1 - 93.1iT - 2.43e4T^{2} \)
31 \( 1 - 111. iT - 2.97e4T^{2} \)
37 \( 1 + 146.T + 5.06e4T^{2} \)
41 \( 1 - 8.44iT - 6.89e4T^{2} \)
43 \( 1 + 427. iT - 7.95e4T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 + 291. iT - 1.48e5T^{2} \)
59 \( 1 - 364.T + 2.05e5T^{2} \)
61 \( 1 + 289.T + 2.26e5T^{2} \)
67 \( 1 + 305. iT - 3.00e5T^{2} \)
71 \( 1 + 102.T + 3.57e5T^{2} \)
73 \( 1 - 442.T + 3.89e5T^{2} \)
79 \( 1 + 245. iT - 4.93e5T^{2} \)
83 \( 1 + 478.T + 5.71e5T^{2} \)
89 \( 1 - 1.41e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−13.49888636816011644603241087877, −12.10615459638430798884517502332, −11.34982912154787092339725455734, −10.52665263586442919075859243535, −9.272728149466355635299710882348, −7.09621761383346818200936900149, −6.65658303165758858182999354030, −5.15760060290614537456062882040, −3.38190806725150980036246672811, −2.40815448797661654392573465023, 1.40338959761437196051778358859, 3.91012189583593365728573490993, 4.79002072264481083695991733866, 6.01534289626603080519119348465, 7.52964128626482945440877209462, 8.582168086721674576702352844996, 9.883130149038244995132864489615, 11.48634514596860123649755214270, 12.40806947656613570717609181852, 13.02896302772665116317571194176

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