Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.996 - 0.0840i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.47 + 2.61i)3-s + (−4.32 − 0.763i)5-s + (−2.73 + 2.29i)7-s + (−4.66 − 7.69i)9-s + (−14.4 + 2.54i)11-s + (−3.67 + 1.33i)13-s + (8.36 − 10.1i)15-s + (−5.96 − 3.44i)17-s + (14.1 + 24.4i)19-s + (−1.97 − 10.5i)21-s + (0.832 − 0.992i)23-s + (−5.33 − 1.94i)25-s + (26.9 − 0.868i)27-s + (−12.9 + 35.7i)29-s + (41.7 + 35.0i)31-s + ⋯
L(s)  = 1  + (−0.490 + 0.871i)3-s + (−0.865 − 0.152i)5-s + (−0.391 + 0.328i)7-s + (−0.518 − 0.855i)9-s + (−1.31 + 0.231i)11-s + (−0.282 + 0.102i)13-s + (0.557 − 0.679i)15-s + (−0.350 − 0.202i)17-s + (0.744 + 1.28i)19-s + (−0.0940 − 0.501i)21-s + (0.0361 − 0.0431i)23-s + (−0.213 − 0.0777i)25-s + (0.999 − 0.0321i)27-s + (−0.448 + 1.23i)29-s + (1.34 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.996 - 0.0840i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (77, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.996 - 0.0840i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0145387 + 0.345230i\)
\(L(\frac12)\)  \(\approx\)  \(0.0145387 + 0.345230i\)
\(L(2)\)   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 \( 1 + (1.47 - 2.61i)T \)
good5 \( 1 + (4.32 + 0.763i)T + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (2.73 - 2.29i)T + (8.50 - 48.2i)T^{2} \)
11 \( 1 + (14.4 - 2.54i)T + (113. - 41.3i)T^{2} \)
13 \( 1 + (3.67 - 1.33i)T + (129. - 108. i)T^{2} \)
17 \( 1 + (5.96 + 3.44i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.1 - 24.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-0.832 + 0.992i)T + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (12.9 - 35.7i)T + (-644. - 540. i)T^{2} \)
31 \( 1 + (-41.7 - 35.0i)T + (166. + 946. i)T^{2} \)
37 \( 1 + (-18.5 + 32.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (14.0 + 38.6i)T + (-1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (0.615 + 3.49i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (27.5 + 32.7i)T + (-383. + 2.17e3i)T^{2} \)
53 \( 1 - 47.8iT - 2.80e3T^{2} \)
59 \( 1 + (61.1 + 10.7i)T + (3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (8.50 - 7.13i)T + (646. - 3.66e3i)T^{2} \)
67 \( 1 + (105. - 38.2i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-90.9 - 52.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (48.5 + 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (104. + 37.9i)T + (4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (44.8 - 123. i)T + (-5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (-87.8 + 50.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (28.7 + 163. i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−14.11879333937456009676597418732, −12.58338140716339609588540769681, −11.92220873945492268029182863681, −10.74459896427661769358677270802, −9.888090663673931339035971299677, −8.641448398984297658819632265850, −7.40308359590694225679718295575, −5.76279788432577722003710088651, −4.64063255257413904733766986368, −3.21843214735252041207298978015, 0.25727940899772211125846840888, 2.80955888739261478059820980298, 4.77045560754975264785443771658, 6.23902872518839100980468121889, 7.47650239163729472367696057323, 8.117615717196629114659774096377, 9.907601150801817684383269565316, 11.18026104418760490929498690574, 11.77632786422860427766631061460, 13.16255930918964871978046585294

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