Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.368 - 0.929i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.67 − 4.43i)5-s + (−30.9 + 53.5i)7-s + (−94.7 − 54.7i)11-s + (77.8 + 134. i)13-s + 395. i·17-s + 140.·19-s + (−802. + 463. i)23-s + (−273. + 473. i)25-s + (−323. − 186. i)29-s + (521. + 903. i)31-s + 548. i·35-s − 194.·37-s + (2.34e3 − 1.35e3i)41-s + (−167. + 290. i)43-s + (−2.46e3 − 1.42e3i)47-s + ⋯
L(s)  = 1  + (0.307 − 0.177i)5-s + (−0.631 + 1.09i)7-s + (−0.783 − 0.452i)11-s + (0.460 + 0.798i)13-s + 1.37i·17-s + 0.388·19-s + (−1.51 + 0.876i)23-s + (−0.437 + 0.757i)25-s + (−0.385 − 0.222i)29-s + (0.543 + 0.940i)31-s + 0.447i·35-s − 0.142·37-s + (1.39 − 0.805i)41-s + (−0.0908 + 0.157i)43-s + (−1.11 − 0.645i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.368 - 0.929i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.368 - 0.929i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.627625 + 0.924119i\)
\(L(\frac12)\)  \(\approx\)  \(0.627625 + 0.924119i\)
\(L(3)\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-7.67 + 4.43i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (30.9 - 53.5i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (94.7 + 54.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-77.8 - 134. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 395. iT - 8.35e4T^{2} \)
19 \( 1 - 140.T + 1.30e5T^{2} \)
23 \( 1 + (802. - 463. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (323. + 186. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-521. - 903. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 194.T + 1.87e6T^{2} \)
41 \( 1 + (-2.34e3 + 1.35e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (167. - 290. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.46e3 + 1.42e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 2.76e3iT - 7.89e6T^{2} \)
59 \( 1 + (-4.35e3 + 2.51e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-3.52e3 + 6.10e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.43e3 + 5.95e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 821. iT - 2.54e7T^{2} \)
73 \( 1 - 4.09e3T + 2.83e7T^{2} \)
79 \( 1 + (3.78e3 - 6.55e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.77e3 + 3.91e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.28e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.89e3 + 3.27e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.23616849041934904196230340103, −12.35443565886840627606288004754, −11.28897455214028263082838600983, −9.998497843946369156977442211746, −9.030843601973789940940111794897, −7.988270068028496779581564907169, −6.28315901136379656944181566204, −5.48849351409636477732708277135, −3.60889813924494450626032073028, −1.96526593375826607732971837729, 0.47976083246582776993304898284, 2.69831798993300103991175674647, 4.25605076254616409835985134904, 5.81875513785665830685523322030, 7.09564378667997127356957082028, 8.096406683699893071194451135252, 9.828443465872066776743382698776, 10.25955067758849588216834170996, 11.56733702747258197163621943672, 12.91070069303591340352701244375

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