Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.206 - 0.978i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 − 0.584i)2-s + (2.24 + 1.98i)3-s + (3.31 + 2.23i)4-s + (0.641 + 3.64i)5-s + (−3.13 − 5.11i)6-s + (−2.33 + 2.78i)7-s + (−5.03 − 6.21i)8-s + (1.08 + 8.93i)9-s + (0.899 − 7.33i)10-s + (−15.6 − 2.76i)11-s + (3.00 + 11.6i)12-s + (13.1 + 4.78i)13-s + (6.09 − 3.96i)14-s + (−5.80 + 9.45i)15-s + (6.00 + 14.8i)16-s + (4.98 + 8.63i)17-s + ⋯
L(s)  = 1  + (−0.956 − 0.292i)2-s + (0.748 + 0.663i)3-s + (0.829 + 0.558i)4-s + (0.128 + 0.728i)5-s + (−0.522 − 0.852i)6-s + (−0.333 + 0.397i)7-s + (−0.629 − 0.776i)8-s + (0.120 + 0.992i)9-s + (0.0899 − 0.733i)10-s + (−1.42 − 0.251i)11-s + (0.250 + 0.968i)12-s + (1.01 + 0.367i)13-s + (0.435 − 0.282i)14-s + (−0.386 + 0.630i)15-s + (0.375 + 0.926i)16-s + (0.293 + 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.206 - 0.978i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.206 - 0.978i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.816917 + 0.662414i\)
\(L(\frac12)\)  \(\approx\)  \(0.816917 + 0.662414i\)
\(L(2)\)   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 \( 1 + (1.91 + 0.584i)T \)
3 \( 1 + (-2.24 - 1.98i)T \)
good5 \( 1 + (-0.641 - 3.64i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (2.33 - 2.78i)T + (-8.50 - 48.2i)T^{2} \)
11 \( 1 + (15.6 + 2.76i)T + (113. + 41.3i)T^{2} \)
13 \( 1 + (-13.1 - 4.78i)T + (129. + 108. i)T^{2} \)
17 \( 1 + (-4.98 - 8.63i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.6 - 8.46i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (0.374 + 0.446i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (-16.6 + 6.04i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (18.3 + 21.8i)T + (-166. + 946. i)T^{2} \)
37 \( 1 + (31.7 + 55.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-59.2 - 21.5i)T + (1.28e3 + 1.08e3i)T^{2} \)
43 \( 1 + (-1.70 - 0.300i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-51.7 + 61.6i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 + 66.4T + 2.80e3T^{2} \)
59 \( 1 + (-65.4 + 11.5i)T + (3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-5.43 - 4.55i)T + (646. + 3.66e3i)T^{2} \)
67 \( 1 + (12.7 - 35.1i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (-12.7 + 7.38i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-66.1 + 114. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (24.4 + 67.1i)T + (-4.78e3 + 4.01e3i)T^{2} \)
83 \( 1 + (-8.25 - 22.6i)T + (-5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (29.6 - 51.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-13.3 + 75.5i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−13.73976261662859161222261711027, −12.59422631182389534622517977474, −11.02208460313660796606563934209, −10.47546448535319067316524446209, −9.445740053926267141193700837401, −8.418410002052440953803658284274, −7.45818493396999781870918505430, −5.85220198238283464566936416724, −3.55675896363506002617186241136, −2.43905948194785939601121678850, 1.02489927911311185568326817667, 2.91661226189439196986513294958, 5.39021091300331919864256874087, 6.89656580168809937672784815920, 7.87377300312316149028435296824, 8.742845404815420626935585720174, 9.744575379408408148052442651859, 10.87062962433356462282949740589, 12.35581249114580813252277724557, 13.25321633625374570522919488528

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