Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $-0.639 - 0.769i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.04 − 3.86i)2-s + (−7.79 + 4.49i)3-s + (−13.8 − 8.03i)4-s + (−5.89 + 10.2i)5-s + (9.23 + 34.7i)6-s + (−50.5 + 29.1i)7-s + (−45.4 + 45.0i)8-s + (40.6 − 70.0i)9-s + (33.2 + 33.3i)10-s + (−86.9 + 50.2i)11-s + (143. + 0.544i)12-s + (85.3 − 147. i)13-s + (60.1 + 225. i)14-s + (0.116 − 106. i)15-s + (126. + 222. i)16-s − 398.·17-s + ⋯
L(s)  = 1  + (0.260 − 0.965i)2-s + (−0.866 + 0.499i)3-s + (−0.864 − 0.502i)4-s + (−0.235 + 0.408i)5-s + (0.256 + 0.966i)6-s + (−1.03 + 0.595i)7-s + (−0.709 + 0.704i)8-s + (0.501 − 0.864i)9-s + (0.332 + 0.333i)10-s + (−0.718 + 0.414i)11-s + (0.999 + 0.00378i)12-s + (0.504 − 0.874i)13-s + (0.306 + 1.15i)14-s + (0.000517 − 0.471i)15-s + (0.495 + 0.868i)16-s − 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.639 - 0.769i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ -0.639 - 0.769i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.0423638 + 0.0902737i\)
\(L(\frac12)\)  \(\approx\)  \(0.0423638 + 0.0902737i\)
\(L(3)\)   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 \( 1 + (-1.04 + 3.86i)T \)
3 \( 1 + (7.79 - 4.49i)T \)
good5 \( 1 + (5.89 - 10.2i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (50.5 - 29.1i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (86.9 - 50.2i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-85.3 + 147. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 398.T + 8.35e4T^{2} \)
19 \( 1 + 404. iT - 1.30e5T^{2} \)
23 \( 1 + (291. + 168. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-327. - 567. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-550. - 317. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 + (1.23e3 - 2.13e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.93e3 + 1.11e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (2.51e3 - 1.45e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.29e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.00e3 - 578. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.96e3 + 5.12e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.08e3 + 1.78e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.49e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.78e3 + 1.61e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (7.06e3 - 4.07e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 910.T + 6.27e7T^{2} \)
97 \( 1 + (-8.80e3 - 1.52e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−15.80643169094520802234854071952, −15.22128231384389707602832122516, −13.26221098432117949590078401150, −12.42649086469288155157249498960, −11.12255862073339188352056805819, −10.28860438517116400705547942737, −9.015426184592627067497984830606, −6.42878490397930157241690141507, −4.91376060083669984152722618892, −3.08633070711112626224748937368, 0.06864290152603763806765706979, 4.24802955128403618833395674596, 5.96711119526864417120526796656, 6.98963174799650032310019804177, 8.461755631021562339728127444019, 10.22605902422810496713046561715, 11.94588558636121481634602959686, 13.11729266800805260499997318745, 13.79191598204031715676331051505, 15.86117611783732742663851345638

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