Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.610 + 0.791i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 1.55i)2-s + (−0.806 + 3.91i)4-s + (−1.35 + 2.34i)5-s + (10.0 − 5.79i)7-s + (7.09 − 3.70i)8-s + (5.35 − 0.865i)10-s + (8.54 − 4.93i)11-s + (0.296 − 0.513i)13-s + (−21.6 − 8.24i)14-s + (−14.6 − 6.31i)16-s + 8.87·17-s − 14.0i·19-s + (−8.10 − 7.20i)20-s + (−18.4 − 7.01i)22-s + (18.2 + 10.5i)23-s + ⋯
L(s)  = 1  + (−0.631 − 0.775i)2-s + (−0.201 + 0.979i)4-s + (−0.271 + 0.469i)5-s + (1.43 − 0.828i)7-s + (0.886 − 0.462i)8-s + (0.535 − 0.0865i)10-s + (0.777 − 0.448i)11-s + (0.0227 − 0.0394i)13-s + (−1.54 − 0.588i)14-s + (−0.918 − 0.394i)16-s + 0.522·17-s − 0.742i·19-s + (−0.405 − 0.360i)20-s + (−0.838 − 0.318i)22-s + (0.794 + 0.458i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.610 + 0.791i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.610 + 0.791i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.987590 - 0.485516i\)
\(L(\frac12)\)  \(\approx\)  \(0.987590 - 0.485516i\)
\(L(2)\)   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 \( 1 + (1.26 + 1.55i)T \)
3 \( 1 \)
good5 \( 1 + (1.35 - 2.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-10.0 + 5.79i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-8.54 + 4.93i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.296 + 0.513i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8.87T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-18.2 - 10.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.1 + 17.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.3 - 8.27i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 40.6T + 1.36e3T^{2} \)
41 \( 1 + (21.2 - 36.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (32.2 - 18.6i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.57 - 0.907i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 21.1T + 2.80e3T^{2} \)
59 \( 1 + (76.6 + 44.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.4 - 63.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (38.3 + 22.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 111. iT - 5.04e3T^{2} \)
73 \( 1 + 76.2T + 5.32e3T^{2} \)
79 \( 1 + (8.30 - 4.79i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (73.6 - 42.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 64.7T + 7.92e3T^{2} \)
97 \( 1 + (3.59 + 6.22i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.29506245696106929012096953107, −11.75186935786872588133994465577, −11.24578278578339375537324254377, −10.37082164571775500307908742920, −9.013414808366224390727828534266, −7.932580486173425985253322414006, −7.00963441947485280818103989243, −4.74210282686623827295023809942, −3.37673861890857071640482653232, −1.32359757136277887153429086983, 1.58243423292837904645637517620, 4.58265542036218223704861650977, 5.59482071114531624799244949933, 7.11173488768734900408786342500, 8.338815806177084170122802843446, 8.880229872104417116596001894549, 10.26358864100356766504868716360, 11.50539383037959967871806778910, 12.39831155175270163498284524754, 14.10475057860647059321427218474

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