Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.369 - 0.929i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.776 + 2.89i)3-s + (2.68 + 3.20i)5-s + (−4.88 + 1.77i)7-s + (−7.79 − 4.49i)9-s + (−4.52 + 5.38i)11-s + (3.33 + 18.8i)13-s + (−11.3 + 5.30i)15-s + (20.3 − 11.7i)17-s + (−11.7 + 20.4i)19-s + (−1.36 − 15.5i)21-s + (8.16 − 22.4i)23-s + (1.30 − 7.41i)25-s + (19.0 − 19.0i)27-s + (21.2 + 3.74i)29-s + (24.0 + 8.73i)31-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (0.537 + 0.640i)5-s + (−0.698 + 0.254i)7-s + (−0.866 − 0.499i)9-s + (−0.411 + 0.489i)11-s + (0.256 + 1.45i)13-s + (−0.757 + 0.353i)15-s + (1.19 − 0.692i)17-s + (−0.620 + 1.07i)19-s + (−0.0648 − 0.739i)21-s + (0.355 − 0.975i)23-s + (0.0523 − 0.296i)25-s + (0.706 − 0.707i)27-s + (0.731 + 0.128i)29-s + (0.774 + 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.369 - 0.929i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (29, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.369 - 0.929i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.641929 + 0.946098i\)
\(L(\frac12)\)  \(\approx\)  \(0.641929 + 0.946098i\)
\(L(2)\)   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 \( 1 \)
3 \( 1 + (0.776 - 2.89i)T \)
good5 \( 1 + (-2.68 - 3.20i)T + (-4.34 + 24.6i)T^{2} \)
7 \( 1 + (4.88 - 1.77i)T + (37.5 - 31.4i)T^{2} \)
11 \( 1 + (4.52 - 5.38i)T + (-21.0 - 119. i)T^{2} \)
13 \( 1 + (-3.33 - 18.8i)T + (-158. + 57.8i)T^{2} \)
17 \( 1 + (-20.3 + 11.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.7 - 20.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.16 + 22.4i)T + (-405. - 340. i)T^{2} \)
29 \( 1 + (-21.2 - 3.74i)T + (790. + 287. i)T^{2} \)
31 \( 1 + (-24.0 - 8.73i)T + (736. + 617. i)T^{2} \)
37 \( 1 + (6.81 + 11.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-50.7 + 8.94i)T + (1.57e3 - 574. i)T^{2} \)
43 \( 1 + (-3.55 - 2.98i)T + (321. + 1.82e3i)T^{2} \)
47 \( 1 + (-1.64 - 4.51i)T + (-1.69e3 + 1.41e3i)T^{2} \)
53 \( 1 + 67.3iT - 2.80e3T^{2} \)
59 \( 1 + (-55.7 - 66.4i)T + (-604. + 3.42e3i)T^{2} \)
61 \( 1 + (-50.3 + 18.3i)T + (2.85e3 - 2.39e3i)T^{2} \)
67 \( 1 + (-3.49 - 19.8i)T + (-4.21e3 + 1.53e3i)T^{2} \)
71 \( 1 + (85.2 - 49.2i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (69.6 - 120. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-18.3 + 103. i)T + (-5.86e3 - 2.13e3i)T^{2} \)
83 \( 1 + (79.2 + 13.9i)T + (6.47e3 + 2.35e3i)T^{2} \)
89 \( 1 + (58.0 + 33.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (112. + 94.0i)T + (1.63e3 + 9.26e3i)T^{2} \)
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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

−14.21149122748211718496999385973, −12.60704195661395666995618354877, −11.61215525034124849500966256055, −10.30585876736991234512251949428, −9.841091912877895943380190568241, −8.646890770936653489118670076730, −6.80772146269679607675021116238, −5.82346298258915850654232765067, −4.32024743543488232197204930985, −2.76194428864353224684456285776, 0.912429131105942198564637914739, 3.01279607383815180428738775410, 5.33436769665136523227374820149, 6.18700892673302925823869822180, 7.61523139299052737000230121376, 8.608191121497711971281083536747, 9.998977283938197638047027497214, 11.07659991952808856715924569879, 12.47621699665746075454089334352, 13.09117377088856626500664249586

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