Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.131 + 0.991i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.157 + 2.82i)2-s + (−7.95 + 0.888i)4-s + (1.23 − 0.715i)5-s + (−23.8 − 13.7i)7-s + (−3.76 − 22.3i)8-s + (2.21 + 3.38i)10-s + (−11.1 + 19.2i)11-s + (−34.5 − 59.9i)13-s + (35.1 − 69.6i)14-s + (62.4 − 14.1i)16-s + 31.4i·17-s − 11.4i·19-s + (−9.21 + 6.78i)20-s + (−56.0 − 28.3i)22-s + (−72.6 − 125. i)23-s + ⋯
L(s)  = 1  + (0.0556 + 0.998i)2-s + (−0.993 + 0.111i)4-s + (0.110 − 0.0639i)5-s + (−1.28 − 0.744i)7-s + (−0.166 − 0.986i)8-s + (0.0700 + 0.107i)10-s + (−0.304 + 0.527i)11-s + (−0.738 − 1.27i)13-s + (0.671 − 1.32i)14-s + (0.975 − 0.220i)16-s + 0.448i·17-s − 0.138i·19-s + (−0.102 + 0.0758i)20-s + (−0.543 − 0.274i)22-s + (−0.658 − 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.131 + 0.991i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.131 + 0.991i)\)
\(L(2)\)  \(\approx\)  \(0.190179 - 0.217028i\)
\(L(\frac12)\)  \(\approx\)  \(0.190179 - 0.217028i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-0.157 - 2.82i)T \)
3 \( 1 \)
good5 \( 1 + (-1.23 + 0.715i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (23.8 + 13.7i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (11.1 - 19.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (34.5 + 59.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 31.4iT - 4.91e3T^{2} \)
19 \( 1 + 11.4iT - 6.85e3T^{2} \)
23 \( 1 + (72.6 + 125. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-93.6 - 54.0i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-102. + 59.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 + (344. - 199. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (173. + 100. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (151. - 262. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 243. iT - 1.48e5T^{2} \)
59 \( 1 + (41.9 + 72.6i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-199. + 345. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-307. + 177. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 866.T + 3.57e5T^{2} \)
73 \( 1 - 64.6T + 3.89e5T^{2} \)
79 \( 1 + (354. + 204. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-79.8 + 138. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + (-700. + 1.21e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.02188769289505187287068971566, −12.44761738121612340857579536417, −10.27753928118727000813453035207, −9.822153650169887624960046862925, −8.327189532647507629103571601392, −7.22981494901524479016281651150, −6.25588308207095518531641743505, −4.88794309829254508587488494321, −3.36256936768755026074541497196, −0.14902179528642485069975522841, 2.28136591066239083552444470907, 3.59427416190026295439793358820, 5.25445497180777568314177267881, 6.60700164991727201804785876578, 8.452609261968922723153605620026, 9.530828351979645655815489616422, 10.15270706698335341716154429589, 11.73304556187499251516824883953, 12.17798478688378572746257334427, 13.44856198211586949383001356639

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