Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.984 + 0.175i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.70 + 1.29i)3-s + (1.65 − 4.54i)5-s + (1.68 − 9.56i)7-s + (5.64 + 7.01i)9-s + (5.23 + 14.3i)11-s + (−6.28 + 5.27i)13-s + (10.3 − 10.1i)15-s + (−9.01 + 5.20i)17-s + (7.17 − 12.4i)19-s + (16.9 − 23.6i)21-s + (−24.2 + 4.27i)23-s + (1.22 + 1.02i)25-s + (6.18 + 26.2i)27-s + (−20.0 + 23.8i)29-s + (−10.5 − 59.6i)31-s + ⋯
L(s)  = 1  + (0.901 + 0.431i)3-s + (0.330 − 0.909i)5-s + (0.240 − 1.36i)7-s + (0.627 + 0.778i)9-s + (0.476 + 1.30i)11-s + (−0.483 + 0.405i)13-s + (0.691 − 0.677i)15-s + (−0.530 + 0.306i)17-s + (0.377 − 0.654i)19-s + (0.807 − 1.12i)21-s + (−1.05 + 0.185i)23-s + (0.0489 + 0.0410i)25-s + (0.229 + 0.973i)27-s + (−0.690 + 0.822i)29-s + (−0.339 − 1.92i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.984 + 0.175i$
motivic weight  =  \(2\)
character  :  $\chi_{108} (65, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :1),\ 0.984 + 0.175i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.84672 - 0.162911i\)
\(L(\frac12)\)  \(\approx\)  \(1.84672 - 0.162911i\)
\(L(2)\)   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 \( 1 + (-2.70 - 1.29i)T \)
good5 \( 1 + (-1.65 + 4.54i)T + (-19.1 - 16.0i)T^{2} \)
7 \( 1 + (-1.68 + 9.56i)T + (-46.0 - 16.7i)T^{2} \)
11 \( 1 + (-5.23 - 14.3i)T + (-92.6 + 77.7i)T^{2} \)
13 \( 1 + (6.28 - 5.27i)T + (29.3 - 166. i)T^{2} \)
17 \( 1 + (9.01 - 5.20i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.17 + 12.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (24.2 - 4.27i)T + (497. - 180. i)T^{2} \)
29 \( 1 + (20.0 - 23.8i)T + (-146. - 828. i)T^{2} \)
31 \( 1 + (10.5 + 59.6i)T + (-903. + 328. i)T^{2} \)
37 \( 1 + (0.367 + 0.636i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-30.1 - 35.9i)T + (-291. + 1.65e3i)T^{2} \)
43 \( 1 + (69.9 - 25.4i)T + (1.41e3 - 1.18e3i)T^{2} \)
47 \( 1 + (12.5 + 2.21i)T + (2.07e3 + 755. i)T^{2} \)
53 \( 1 - 36.5iT - 2.80e3T^{2} \)
59 \( 1 + (-30.5 + 83.8i)T + (-2.66e3 - 2.23e3i)T^{2} \)
61 \( 1 + (7.06 - 40.0i)T + (-3.49e3 - 1.27e3i)T^{2} \)
67 \( 1 + (-61.6 + 51.6i)T + (779. - 4.42e3i)T^{2} \)
71 \( 1 + (-0.595 + 0.343i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-13.7 + 23.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-97.1 - 81.5i)T + (1.08e3 + 6.14e3i)T^{2} \)
83 \( 1 + (8.62 - 10.2i)T + (-1.19e3 - 6.78e3i)T^{2} \)
89 \( 1 + (146. + 84.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (60.2 - 21.9i)T + (7.20e3 - 6.04e3i)T^{2} \)
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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

−13.46379432619163715745712420472, −12.74221488910465536996684470863, −11.21569379978909463605226145004, −9.853643394175196125243010727506, −9.371075792217267026515745145089, −7.968910509739457440883385289932, −7.00973043650395209652173509276, −4.81472505187796052668009582317, −4.04251258786115456042337624902, −1.79153527413412053236722385613, 2.22913986770602028501236805863, 3.37753677183457190467619473882, 5.67288868312043965198120885383, 6.76522207875571026594253001249, 8.185601743841184956417768096390, 8.984315312003144455324221813707, 10.17975507142981790815009011792, 11.56013244875637367150412022823, 12.45320126017552681876459419560, 13.78205080247505724683732020598

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