Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.999 + 0.0422i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.523 − 1.93i)2-s + (−3.45 + 2.02i)4-s + (4.03 + 6.98i)5-s + (3.90 + 2.25i)7-s + (5.71 + 5.60i)8-s + (11.3 − 11.4i)10-s + (3.25 + 1.88i)11-s + (−3.52 − 6.10i)13-s + (2.30 − 8.71i)14-s + (7.81 − 13.9i)16-s − 0.517·17-s + 16.4i·19-s + (−28.0 − 15.9i)20-s + (1.92 − 7.27i)22-s + (27.7 − 15.9i)23-s + ⋯
L(s)  = 1  + (−0.261 − 0.965i)2-s + (−0.862 + 0.505i)4-s + (0.806 + 1.39i)5-s + (0.557 + 0.321i)7-s + (0.713 + 0.700i)8-s + (1.13 − 1.14i)10-s + (0.296 + 0.171i)11-s + (−0.271 − 0.469i)13-s + (0.164 − 0.622i)14-s + (0.488 − 0.872i)16-s − 0.0304·17-s + 0.864i·19-s + (−1.40 − 0.797i)20-s + (0.0874 − 0.330i)22-s + (1.20 − 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0422i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.999 + 0.0422i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.999 + 0.0422i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.28073 - 0.0270693i\)
\(L(\frac12)\)  \(\approx\)  \(1.28073 - 0.0270693i\)
\(L(2)\)   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 \( 1 + (0.523 + 1.93i)T \)
3 \( 1 \)
good5 \( 1 + (-4.03 - 6.98i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.90 - 2.25i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.25 - 1.88i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.52 + 6.10i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 0.517T + 289T^{2} \)
19 \( 1 - 16.4iT - 361T^{2} \)
23 \( 1 + (-27.7 + 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (9.48 - 16.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-13.1 + 7.58i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 0.592T + 1.36e3T^{2} \)
41 \( 1 + (12.3 + 21.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (27.8 + 16.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (52.4 + 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 0.664T + 2.80e3T^{2} \)
59 \( 1 + (30.5 - 17.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-33.7 + 58.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-74.4 + 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 56.4iT - 5.04e3T^{2} \)
73 \( 1 - 131.T + 5.32e3T^{2} \)
79 \( 1 + (126. + 73.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-87.1 - 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 25.8T + 7.92e3T^{2} \)
97 \( 1 + (48.2 - 83.5i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.40403578536495006056176323677, −12.25155621458818274238017771132, −11.13562173508544334610190444773, −10.41492524851464640594590099389, −9.506998579739736874976639770477, −8.171503262092815779821588127421, −6.77663155055773164266342132407, −5.20017798648788295337234890206, −3.31536556979562851607850691726, −2.02593455528903372581063402357, 1.23891170162293392259847744849, 4.53893743588773792954033295859, 5.34232198039065833827979255929, 6.71693150288283800084612902597, 8.098408444905022745344675660051, 9.067593666206437326338883729019, 9.763475489549978361521105626373, 11.34091721141724770953705324120, 12.87553302159517635197684909033, 13.53612027589168671800754716211

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