Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.998 + 0.0552i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.93 − 0.511i)2-s + (3.47 − 1.97i)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 + (−8.69 − 2.36i)14-s + (8.17 − 13.7i)16-s − 0.517·17-s − 16.4i·19-s + (27.8 + 16.3i)20-s + (−7.26 − 1.97i)22-s + (−27.7 + 15.9i)23-s + ⋯
L(s)  = 1  + (0.966 − 0.255i)2-s + (0.869 − 0.494i)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 + (−0.621 − 0.168i)14-s + (0.511 − 0.859i)16-s − 0.0304·17-s − 0.864i·19-s + (1.39 + 0.815i)20-s + (−0.330 − 0.0896i)22-s + (−1.20 + 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.998 + 0.0552i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.998 + 0.0552i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.41215 - 0.0667008i\)
\(L(\frac12)\)  \(\approx\)  \(2.41215 - 0.0667008i\)
\(L(2)\)   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 \( 1 + (-1.93 + 0.511i)T \)
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

−13.60310051357388997575180864599, −12.64360442745443232132112530829, −11.26146812701962823556462750772, −10.47894248944028022161083930517, −9.670381083623423451387667227975, −7.41423833244228633778436216296, −6.50503903843277298688683742791, −5.46812120106627232328180055858, −3.56555384082555777691372785635, −2.41419398628813141766989360955, 2.11785979711276322974283643624, 4.16807801295141404521366632360, 5.39764085965707006277029392572, 6.26768008164503028860761021338, 7.900386126115795482313095759688, 9.125087980184593831932152797275, 10.25723566422469461473863379247, 11.95765935204822006584263752241, 12.56942316592647186553620841738, 13.39436312607062602640502414258

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