Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.854 + 0.519i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.14 − 3.37i)2-s + (−6.83 − 14.4i)4-s + (14.8 + 25.7i)5-s + (51.8 + 29.9i)7-s + (−63.5 − 7.86i)8-s + (118. + 4.89i)10-s + (195. + 112. i)11-s + (−85.8 − 148. i)13-s + (212. − 111. i)14-s + (−162. + 197. i)16-s + 99.0·17-s − 169. i·19-s + (271. − 391. i)20-s + (798. − 418. i)22-s + (310. − 179. i)23-s + ⋯
L(s)  = 1  + (0.535 − 0.844i)2-s + (−0.427 − 0.904i)4-s + (0.595 + 1.03i)5-s + (1.05 + 0.611i)7-s + (−0.992 − 0.122i)8-s + (1.18 + 0.0489i)10-s + (1.61 + 0.931i)11-s + (−0.508 − 0.880i)13-s + (1.08 − 0.567i)14-s + (−0.634 + 0.772i)16-s + 0.342·17-s − 0.468i·19-s + (0.678 − 0.978i)20-s + (1.65 − 0.864i)22-s + (0.587 − 0.339i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.854 + 0.519i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.854 + 0.519i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.62450 - 0.734946i\)
\(L(\frac12)\)  \(\approx\)  \(2.62450 - 0.734946i\)
\(L(3)\)   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 \( 1 + (-2.14 + 3.37i)T \)
3 \( 1 \)
good5 \( 1 + (-14.8 - 25.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-51.8 - 29.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-195. - 112. i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (85.8 + 148. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 99.0T + 8.35e4T^{2} \)
19 \( 1 + 169. iT - 1.30e5T^{2} \)
23 \( 1 + (-310. + 179. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (9.01 - 15.6i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (671. - 387. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 609.T + 1.87e6T^{2} \)
41 \( 1 + (206. + 357. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (265. + 153. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.27e3 - 1.31e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 2.03e3T + 7.89e6T^{2} \)
59 \( 1 + (2.25e3 - 1.29e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-708. + 1.22e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (5.19e3 - 2.99e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.23e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.06e3T + 2.83e7T^{2} \)
79 \( 1 + (1.63e3 + 944. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (5.83e3 + 3.36e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 9.43e3T + 6.27e7T^{2} \)
97 \( 1 + (-7.29e3 + 1.26e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.70804686268183247468009200811, −11.82518378058633606400503730770, −10.92439269123915850480355516903, −9.938756052292613423074265874324, −8.914380876252165186598863945939, −7.08341749912145939491660694783, −5.80463904640706733934561793002, −4.54064008422665841144659909778, −2.85766829686020578950426056583, −1.61198891352624046655382424980, 1.32183686428510433684146072587, 3.92941647796895018689836792069, 4.96037420676422702032505744726, 6.13780652508637198313641756842, 7.45531456315711950295264423297, 8.688597628877028946233783441673, 9.390075634820512088985958136440, 11.32712579457480505022756202541, 12.17506422687715013223800941769, 13.39814298502985149887116684673

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