Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.0482 - 0.998i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.99 − 0.277i)2-s + (15.8 + 2.21i)4-s + (23.3 + 40.4i)5-s + (52.4 + 30.2i)7-s + (−62.6 − 13.2i)8-s + (−81.9 − 167. i)10-s + (−63.7 − 36.8i)11-s + (15.5 + 27.0i)13-s + (−200. − 135. i)14-s + (246. + 70.2i)16-s − 53.8·17-s + 54.9i·19-s + (280. + 692. i)20-s + (244. + 164. i)22-s + (243. − 140. i)23-s + ⋯
L(s)  = 1  + (−0.997 − 0.0694i)2-s + (0.990 + 0.138i)4-s + (0.933 + 1.61i)5-s + (1.07 + 0.617i)7-s + (−0.978 − 0.206i)8-s + (−0.819 − 1.67i)10-s + (−0.526 − 0.304i)11-s + (0.0922 + 0.159i)13-s + (−1.02 − 0.690i)14-s + (0.961 + 0.274i)16-s − 0.186·17-s + 0.152i·19-s + (0.700 + 1.73i)20-s + (0.504 + 0.340i)22-s + (0.460 − 0.266i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.0482 - 0.998i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.0482 - 0.998i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.919527 + 0.965063i\)
\(L(\frac12)\)  \(\approx\)  \(0.919527 + 0.965063i\)
\(L(3)\)   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 \( 1 + (3.99 + 0.277i)T \)
3 \( 1 \)
good5 \( 1 + (-23.3 - 40.4i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-52.4 - 30.2i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (63.7 + 36.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-15.5 - 27.0i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 53.8T + 8.35e4T^{2} \)
19 \( 1 - 54.9iT - 1.30e5T^{2} \)
23 \( 1 + (-243. + 140. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (223. - 387. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (240. - 138. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 1.01e3T + 1.87e6T^{2} \)
41 \( 1 + (-946. - 1.63e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (666. + 384. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (2.37e3 + 1.37e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 4.64e3T + 7.89e6T^{2} \)
59 \( 1 + (-262. + 151. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (478. - 828. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (6.01e3 - 3.47e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 5.97e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.33e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.29e3 - 1.90e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-2.73e3 - 1.57e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 7.13e3T + 6.27e7T^{2} \)
97 \( 1 + (-980. + 1.69e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.34206381158078930625388502030, −11.68150197490339002741727569649, −10.94022054328130616523366865062, −10.19532694592613799037283399553, −9.011276557983579214032530376917, −7.79944819512772254240517688396, −6.68924324471181576820233543936, −5.60413710046434894987950960238, −2.93113491312130298394450549633, −1.87310219493381258239741080215, 0.838372344604140998672076066463, 1.97143085286646495686567277294, 4.70349417415704436144232916327, 5.80123432200859524095148366715, 7.52568574984634361978871518782, 8.462113373267362033749291787820, 9.366493727465892029379009597885, 10.36220997316482104476601590684, 11.49049890837447814054323606221, 12.69503533441001684530462801685

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