Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.0796 + 0.996i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 − 1.03i)2-s + (5.16 + 0.555i)3-s + (5.86 + 5.43i)4-s + (6.86 − 18.8i)5-s + (−13.0 − 6.79i)6-s + (−18.0 − 3.17i)7-s + (−9.82 − 20.3i)8-s + (26.3 + 5.73i)9-s + (−37.5 + 42.5i)10-s + (−8.64 + 3.14i)11-s + (27.2 + 31.3i)12-s + (23.2 − 19.4i)13-s + (44.1 + 26.9i)14-s + (45.9 − 93.6i)15-s + (4.82 + 63.8i)16-s + (26.8 − 15.4i)17-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + (0.994 + 0.106i)3-s + (0.733 + 0.679i)4-s + (0.614 − 1.68i)5-s + (−0.886 − 0.462i)6-s + (−0.972 − 0.171i)7-s + (−0.434 − 0.900i)8-s + (0.977 + 0.212i)9-s + (−1.18 + 1.34i)10-s + (−0.237 + 0.0862i)11-s + (0.656 + 0.754i)12-s + (0.495 − 0.415i)13-s + (0.842 + 0.514i)14-s + (0.790 − 1.61i)15-s + (0.0754 + 0.997i)16-s + (0.382 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0796 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0796 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.0796 + 0.996i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.0796 + 0.996i)\)
\(L(2)\)  \(\approx\)  \(0.950910 - 1.02996i\)
\(L(\frac12)\)  \(\approx\)  \(0.950910 - 1.02996i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.63 + 1.03i)T \)
3 \( 1 + (-5.16 - 0.555i)T \)
good5 \( 1 + (-6.86 + 18.8i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (18.0 + 3.17i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (8.64 - 3.14i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-23.2 + 19.4i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-26.8 + 15.4i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (113. + 65.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-9.01 - 51.1i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-166. + 198. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-176. + 31.0i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-134. - 232. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (36.5 + 43.5i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-120. - 331. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-41.3 + 234. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 - 424. iT - 1.48e5T^{2} \)
59 \( 1 + (79.6 + 28.9i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (143. - 813. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (-328. - 390. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-37.9 - 65.6i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (340. - 589. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-756. + 901. i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-163. - 136. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (-719. - 415. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-571. + 207. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−13.16136778985452533599499312636, −12.08115447269212362145814973306, −10.26805529675745415450136517584, −9.583724209563777019369547906758, −8.740091454865314790171967934702, −7.984873674253504117407975436996, −6.33284310712940736140326315849, −4.34647293528992680715443823691, −2.64084966500236073458448170820, −0.963316385047051414469324795133, 2.16513653678960378689794739410, 3.27961239148486963515328691511, 6.28319184113022120021792837483, 6.74975134590521760418342525094, 8.060763644740017972508944529661, 9.233227996792134505317024259761, 10.20775938952040043619316803299, 10.75544099871270988555185366365, 12.54963340881943047649128686463, 13.95675525718761661050627865667

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