Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.955 - 0.293i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.419 − 2.79i)2-s + (−7.64 − 2.34i)4-s + 1.49i·5-s − 26.1i·7-s + (−9.78 + 20.4i)8-s + (4.18 + 0.627i)10-s − 56.3·11-s − 41.3·13-s + (−73.2 − 10.9i)14-s + (52.9 + 35.9i)16-s + 51.0i·17-s − 79.0i·19-s + (3.51 − 11.4i)20-s + (−23.6 + 157. i)22-s − 27.3·23-s + ⋯
L(s)  = 1  + (0.148 − 0.988i)2-s + (−0.955 − 0.293i)4-s + 0.133i·5-s − 1.41i·7-s + (−0.432 + 0.901i)8-s + (0.132 + 0.0198i)10-s − 1.54·11-s − 0.881·13-s + (−1.39 − 0.209i)14-s + (0.827 + 0.561i)16-s + 0.728i·17-s − 0.954i·19-s + (0.0392 − 0.127i)20-s + (−0.229 + 1.52i)22-s − 0.248·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.955 - 0.293i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.955 - 0.293i)\)
\(L(2)\)  \(\approx\)  \(0.117878 + 0.785220i\)
\(L(\frac12)\)  \(\approx\)  \(0.117878 + 0.785220i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-0.419 + 2.79i)T \)
3 \( 1 \)
good5 \( 1 - 1.49iT - 125T^{2} \)
7 \( 1 + 26.1iT - 343T^{2} \)
11 \( 1 + 56.3T + 1.33e3T^{2} \)
13 \( 1 + 41.3T + 2.19e3T^{2} \)
17 \( 1 - 51.0iT - 4.91e3T^{2} \)
19 \( 1 + 79.0iT - 6.85e3T^{2} \)
23 \( 1 + 27.3T + 1.21e4T^{2} \)
29 \( 1 + 134. iT - 2.43e4T^{2} \)
31 \( 1 + 187. iT - 2.97e4T^{2} \)
37 \( 1 + 196.T + 5.06e4T^{2} \)
41 \( 1 - 298. iT - 6.89e4T^{2} \)
43 \( 1 + 465. iT - 7.95e4T^{2} \)
47 \( 1 - 373.T + 1.03e5T^{2} \)
53 \( 1 + 620. iT - 1.48e5T^{2} \)
59 \( 1 - 321.T + 2.05e5T^{2} \)
61 \( 1 - 674.T + 2.26e5T^{2} \)
67 \( 1 - 576. iT - 3.00e5T^{2} \)
71 \( 1 + 223.T + 3.57e5T^{2} \)
73 \( 1 - 70.1T + 3.89e5T^{2} \)
79 \( 1 - 1.05e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + 1.34e3iT - 7.04e5T^{2} \)
97 \( 1 + 576.T + 9.12e5T^{2} \)
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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

−12.84918884044671511239894361725, −11.44062131979484518613368177852, −10.47127404205020536756786962905, −9.953928918539182458067417306933, −8.336995056567058401832769992642, −7.17230752786748737743345569730, −5.25765727952813665531752714032, −4.07148051659690706220469174416, −2.51266461732731790883521016926, −0.40363863507521334178840110270, 2.82343097697133271625410694667, 4.97360446399956787010489099854, 5.63380386291959675188505703991, 7.16788307045704653503900155215, 8.273063835946442654240795925046, 9.188287060584829001047179541825, 10.39989383955569810216331659017, 12.22020383739649237705364211551, 12.67396630576749896379312081209, 14.00901541418725723207767473108

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