Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.748 + 0.663i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (55.1 − 95.6i)5-s + (−50.8 − 88.1i)7-s + (−75.1 − 130. i)11-s + (−317. + 550. i)13-s − 1.49e3·17-s + 1.43e3·19-s + (−632. + 1.09e3i)23-s + (−4.53e3 − 7.84e3i)25-s + (−1.38e3 − 2.40e3i)29-s + (3.48e3 − 6.03e3i)31-s − 1.12e4·35-s − 7.95e3·37-s + (−1.01e3 + 1.75e3i)41-s + (6.26e3 + 1.08e4i)43-s + (−3.24e3 − 5.61e3i)47-s + ⋯
L(s)  = 1  + (0.987 − 1.71i)5-s + (−0.392 − 0.679i)7-s + (−0.187 − 0.324i)11-s + (−0.521 + 0.903i)13-s − 1.25·17-s + 0.913·19-s + (−0.249 + 0.431i)23-s + (−1.45 − 2.51i)25-s + (−0.306 − 0.531i)29-s + (0.651 − 1.12i)31-s − 1.54·35-s − 0.954·37-s + (−0.0941 + 0.163i)41-s + (0.516 + 0.894i)43-s + (−0.214 − 0.370i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.748 + 0.663i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.748 + 0.663i)\)
\(L(3)\)  \(\approx\)  \(0.530684 - 1.39821i\)
\(L(\frac12)\)  \(\approx\)  \(0.530684 - 1.39821i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-55.1 + 95.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (50.8 + 88.1i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (75.1 + 130. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (317. - 550. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 1.49e3T + 1.41e6T^{2} \)
19 \( 1 - 1.43e3T + 2.47e6T^{2} \)
23 \( 1 + (632. - 1.09e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.38e3 + 2.40e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-3.48e3 + 6.03e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 7.95e3T + 6.93e7T^{2} \)
41 \( 1 + (1.01e3 - 1.75e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-6.26e3 - 1.08e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (3.24e3 + 5.61e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 9.82e3T + 4.18e8T^{2} \)
59 \( 1 + (-2.35e4 + 4.07e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.16e3 + 7.22e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-3.63e3 + 6.28e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.58e3T + 1.80e9T^{2} \)
73 \( 1 - 5.80e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.18e4 - 5.52e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.14e4 - 7.17e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 3.86e3T + 5.58e9T^{2} \)
97 \( 1 + (3.46e4 + 5.99e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.56071238751247465274574023764, −11.42546951728256353740785635741, −9.830283887125335764302287468261, −9.311932407493609352093070330283, −8.126261027688820637606489152982, −6.56102353447260575270790731081, −5.28487849199109941789765398824, −4.22142387235296441314776340950, −1.98470357542313506863947734377, −0.53550825493019826634703047970, 2.24116508133253446554311065515, 3.13950726285131638840487566065, 5.36112478217464249112023243961, 6.45142048815562338660771355624, 7.35310348876182769208335852246, 9.069691033098694739444807683490, 10.14654332288907713887870643961, 10.78051424695028313329115760192, 12.11324118161796998657664479781, 13.31199708218420333620298723939

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