Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.191 - 0.981i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.98 + 0.385i)2-s + (15.7 + 3.06i)4-s − 22.8·5-s + 82.1i·7-s + (61.3 + 18.2i)8-s + (−90.8 − 8.79i)10-s + 107. i·11-s + 57.1·13-s + (−31.6 + 326. i)14-s + (237. + 96.3i)16-s + 291.·17-s + 304. i·19-s + (−358. − 69.9i)20-s + (−41.3 + 427. i)22-s − 982. i·23-s + ⋯
L(s)  = 1  + (0.995 + 0.0963i)2-s + (0.981 + 0.191i)4-s − 0.912·5-s + 1.67i·7-s + (0.958 + 0.285i)8-s + (−0.908 − 0.0879i)10-s + 0.886i·11-s + 0.338·13-s + (−0.161 + 1.66i)14-s + (0.926 + 0.376i)16-s + 1.01·17-s + 0.844i·19-s + (−0.895 − 0.174i)20-s + (−0.0854 + 0.882i)22-s − 1.85i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.191 - 0.981i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.191 - 0.981i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.11041 + 1.73791i\)
\(L(\frac12)\)  \(\approx\)  \(2.11041 + 1.73791i\)
\(L(3)\)   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 \( 1 + (-3.98 - 0.385i)T \)
3 \( 1 \)
good5 \( 1 + 22.8T + 625T^{2} \)
7 \( 1 - 82.1iT - 2.40e3T^{2} \)
11 \( 1 - 107. iT - 1.46e4T^{2} \)
13 \( 1 - 57.1T + 2.85e4T^{2} \)
17 \( 1 - 291.T + 8.35e4T^{2} \)
19 \( 1 - 304. iT - 1.30e5T^{2} \)
23 \( 1 + 982. iT - 2.79e5T^{2} \)
29 \( 1 + 1.04e3T + 7.07e5T^{2} \)
31 \( 1 - 783. iT - 9.23e5T^{2} \)
37 \( 1 - 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + 363.T + 2.82e6T^{2} \)
43 \( 1 + 64.7iT - 3.41e6T^{2} \)
47 \( 1 + 3.49e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.26e3T + 7.89e6T^{2} \)
59 \( 1 - 867. iT - 1.21e7T^{2} \)
61 \( 1 - 6.03e3T + 1.38e7T^{2} \)
67 \( 1 + 3.42e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.18e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.26e3T + 2.83e7T^{2} \)
79 \( 1 + 7.53e3iT - 3.89e7T^{2} \)
83 \( 1 - 586. iT - 4.74e7T^{2} \)
89 \( 1 - 5.78e3T + 6.27e7T^{2} \)
97 \( 1 - 1.08e4T + 8.85e7T^{2} \)
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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

−12.93346047013402664455968144131, −12.15932412135813015088854440279, −11.69596579797164615393301471896, −10.25198419688874571096874410218, −8.628598370931448421298544677342, −7.58773217893482780487292918149, −6.18054286805250969721957010538, −5.08024114180216308276967252368, −3.68640462718529265663167865045, −2.21436778448499811793107552268, 0.942245181934765983894894160047, 3.42752029894637123707944932934, 4.12838794455707207553883913713, 5.70288656347808523271993294232, 7.22743928685318371889043906222, 7.83257310886996592964145201714, 9.862641830690986508013137314410, 11.18291982891348338473155450033, 11.46421022188847847569769619748, 13.06668296078238771587071414450

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