Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.547 + 0.836i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.55 − 3.78i)5-s + (−4.55 − 7.89i)7-s + (0.383 − 0.221i)11-s + (5.55 − 9.62i)13-s + 8.01i·17-s − 8.11·19-s + (−20.4 − 11.8i)23-s + (16.1 + 28.0i)25-s + (45.9 − 26.5i)29-s + (−14.6 + 25.4i)31-s + 69.0i·35-s + 18.4·37-s + (38.9 + 22.4i)41-s + (−11.5 − 19.9i)43-s + (7.32 − 4.22i)47-s + ⋯
L(s)  = 1  + (−1.31 − 0.757i)5-s + (−0.651 − 1.12i)7-s + (0.0348 − 0.0201i)11-s + (0.427 − 0.740i)13-s + 0.471i·17-s − 0.427·19-s + (−0.888 − 0.513i)23-s + (0.647 + 1.12i)25-s + (1.58 − 0.913i)29-s + (−0.473 + 0.819i)31-s + 1.97i·35-s + 0.499·37-s + (0.950 + 0.548i)41-s + (−0.267 − 0.463i)43-s + (0.155 − 0.0899i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.547 + 0.836i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ -0.547 + 0.836i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.350911 - 0.649285i\)
\(L(\frac12)\)  \(\approx\)  \(0.350911 - 0.649285i\)
\(L(2)\)   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 \( 1 \)
good5 \( 1 + (6.55 + 3.78i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.55 + 7.89i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.383 + 0.221i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.55 + 9.62i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8.01iT - 289T^{2} \)
19 \( 1 + 8.11T + 361T^{2} \)
23 \( 1 + (20.4 + 11.8i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-45.9 + 26.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (14.6 - 25.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 18.4T + 1.36e3T^{2} \)
41 \( 1 + (-38.9 - 22.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-7.32 + 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 60.5iT - 2.80e3T^{2} \)
59 \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.67 + 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.8 + 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.0iT - 5.04e3T^{2} \)
73 \( 1 + 4.35T + 5.32e3T^{2} \)
79 \( 1 + (0.792 + 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-7.32 + 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 64.1iT - 7.92e3T^{2} \)
97 \( 1 + (57.6 + 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.93342406873095847022393732763, −12.23183158276761048839604988082, −10.98859339043037774353796527527, −10.03214952058346410208112793637, −8.489384269540464530431335121487, −7.74536939872732157704263464981, −6.39697447330836292472837099646, −4.53918976005238821994625486528, −3.57298324427366690752001843619, −0.54625460819383195273545340598, 2.81150799542191288782567888909, 4.15995692806504865553394132840, 6.02350480156748000235432463639, 7.14302323102791064878180342171, 8.362972879649692017780694826245, 9.457149691829423465927597262690, 10.88192292802850518275833364955, 11.78861517075907107398606065031, 12.48027361426847536681996231335, 13.94513067956433929254698145040

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