Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.928 - 0.371i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.33 + 8.69i)3-s + (15.1 + 41.4i)5-s + (8.53 + 48.4i)7-s + (−70.0 − 40.6i)9-s + (43.7 − 120. i)11-s + (6.33 + 5.31i)13-s + (−395. + 34.3i)15-s + (51.3 + 29.6i)17-s + (195. + 337. i)19-s + (−440. − 38.9i)21-s + (−956. − 168. i)23-s + (−1.01e3 + 851. i)25-s + (516. − 514. i)27-s + (342. + 408. i)29-s + (267. − 1.51e3i)31-s + ⋯
L(s)  = 1  + (−0.259 + 0.965i)3-s + (0.604 + 1.65i)5-s + (0.174 + 0.988i)7-s + (−0.865 − 0.501i)9-s + (0.361 − 0.992i)11-s + (0.0374 + 0.0314i)13-s + (−1.75 + 0.152i)15-s + (0.177 + 0.102i)17-s + (0.540 + 0.935i)19-s + (−0.999 − 0.0882i)21-s + (−1.80 − 0.318i)23-s + (−1.62 + 1.36i)25-s + (0.708 − 0.705i)27-s + (0.407 + 0.485i)29-s + (0.278 − 1.57i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.928 - 0.371i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.928 - 0.371i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.287569 + 1.49338i\)
\(L(\frac12)\)  \(\approx\)  \(0.287569 + 1.49338i\)
\(L(3)\)   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 \( 1 + (2.33 - 8.69i)T \)
good5 \( 1 + (-15.1 - 41.4i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (-8.53 - 48.4i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (-43.7 + 120. i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (-6.33 - 5.31i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (-51.3 - 29.6i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-195. - 337. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (956. + 168. i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (-342. - 408. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-267. + 1.51e3i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (-157. + 272. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (1.52e3 - 1.81e3i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (-2.05e3 - 749. i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-1.94e3 + 342. i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 - 2.83e3iT - 7.89e6T^{2} \)
59 \( 1 + (1.44e3 + 3.95e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (170. + 969. i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (-3.05e3 - 2.55e3i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (2.35e3 + 1.36e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-4.48e3 - 7.76e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.15e3 + 1.80e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (1.37e3 + 1.63e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (925. - 534. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (-1.53e4 - 5.58e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−13.94626480271485858690032289857, −12.00459387496449251851746904700, −11.22294805317268272005905809065, −10.27735689032132669534245486386, −9.466397833325766484415186049903, −8.091994334493527082629365278034, −6.22654743476434499144143878392, −5.75801488280951723210002945140, −3.72712266496399656208147690244, −2.53220542826184879420298033195, 0.71412074607573921643683358335, 1.81520699630789104993132972674, 4.42987307789846033626882130574, 5.53465714585539806131892967332, 6.95828594472664161725916760554, 8.046244625221056912570337921321, 9.178321486502520078550964781016, 10.33220396394285779599547942711, 11.95768969169009121527929117719, 12.44896940254481140413137310537

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