Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.749 + 0.662i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.6 + 6.14i)5-s + (7.14 − 12.3i)7-s + (−90.2 − 52.0i)11-s + (−37.6 − 65.1i)13-s − 341. i·17-s − 706.·19-s + (516. − 298. i)23-s + (−237. + 410. i)25-s + (−1.12e3 − 651. i)29-s + (−514. − 891. i)31-s + 175. i·35-s + 563.·37-s + (−85.8 + 49.5i)41-s + (448. − 776. i)43-s + (−372. − 215. i)47-s + ⋯
L(s)  = 1  + (−0.425 + 0.245i)5-s + (0.145 − 0.252i)7-s + (−0.745 − 0.430i)11-s + (−0.222 − 0.385i)13-s − 1.18i·17-s − 1.95·19-s + (0.976 − 0.563i)23-s + (−0.379 + 0.657i)25-s + (−1.34 − 0.774i)29-s + (−0.535 − 0.927i)31-s + 0.143i·35-s + 0.411·37-s + (−0.0510 + 0.0294i)41-s + (0.242 − 0.419i)43-s + (−0.168 − 0.0974i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.749 + 0.662i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (89, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.749 + 0.662i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.211921 - 0.559470i\)
\(L(\frac12)\)  \(\approx\)  \(0.211921 - 0.559470i\)
\(L(3)\)   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 \( 1 \)
good5 \( 1 + (10.6 - 6.14i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-7.14 + 12.3i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (90.2 + 52.0i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (37.6 + 65.1i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 341. iT - 8.35e4T^{2} \)
19 \( 1 + 706.T + 1.30e5T^{2} \)
23 \( 1 + (-516. + 298. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.12e3 + 651. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (514. + 891. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 563.T + 1.87e6T^{2} \)
41 \( 1 + (85.8 - 49.5i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-448. + 776. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (372. + 215. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 5.27e3iT - 7.89e6T^{2} \)
59 \( 1 + (-4.88e3 + 2.81e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (565. - 979. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-676. - 1.17e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 5.68e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.23e3T + 2.83e7T^{2} \)
79 \( 1 + (-3.06e3 + 5.31e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (6.50e3 + 3.75e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 8.72e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.72e3 - 4.71e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.71488456433569075815578670651, −11.32655243337715812596621894530, −10.69036188578142727889393103710, −9.330623820302314809490041364441, −8.065043980709796262134856907268, −7.10884661743073018683087336789, −5.61060217594513426481135999985, −4.17445744714903414580955188443, −2.57642796736828470645343009869, −0.25181078929944249186948372453, 2.02686155251008694596792019030, 3.93077098241084770941989653096, 5.23363382367211402946438630424, 6.71060114834652450625929082702, 8.013861801341204167727120487863, 8.931108773078019095731227093512, 10.33081129273720379339018965693, 11.24383880767117004219228915309, 12.54926106192211155431958350472, 13.06884891300781696216993948461

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