Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.359 + 0.933i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.10 − 1.88i)2-s + (−3.74 − 3.60i)3-s + (0.868 + 7.95i)4-s + (−1.60 + 4.41i)5-s + (1.07 + 14.6i)6-s + (23.1 + 4.07i)7-s + (13.1 − 18.3i)8-s + (1.00 + 26.9i)9-s + (11.7 − 6.25i)10-s + (−26.1 + 9.51i)11-s + (25.4 − 32.8i)12-s + (58.1 − 48.8i)13-s + (−41.0 − 52.2i)14-s + (21.9 − 10.7i)15-s + (−62.4 + 13.8i)16-s + (21.6 − 12.5i)17-s + ⋯
L(s)  = 1  + (−0.744 − 0.667i)2-s + (−0.720 − 0.693i)3-s + (0.108 + 0.994i)4-s + (−0.143 + 0.394i)5-s + (0.0728 + 0.997i)6-s + (1.24 + 0.220i)7-s + (0.582 − 0.812i)8-s + (0.0370 + 0.999i)9-s + (0.370 − 0.197i)10-s + (−0.716 + 0.260i)11-s + (0.611 − 0.791i)12-s + (1.24 − 1.04i)13-s + (−0.782 − 0.997i)14-s + (0.377 − 0.184i)15-s + (−0.976 + 0.215i)16-s + (0.309 − 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.359 + 0.933i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ 0.359 + 0.933i)\)
\(L(2)\)  \(\approx\)  \(0.777944 - 0.533821i\)
\(L(\frac12)\)  \(\approx\)  \(0.777944 - 0.533821i\)
\(L(\frac{5}{2})\)   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 \( 1 + (2.10 + 1.88i)T \)
3 \( 1 + (3.74 + 3.60i)T \)
good5 \( 1 + (1.60 - 4.41i)T + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (-23.1 - 4.07i)T + (322. + 117. i)T^{2} \)
11 \( 1 + (26.1 - 9.51i)T + (1.01e3 - 855. i)T^{2} \)
13 \( 1 + (-58.1 + 48.8i)T + (381. - 2.16e3i)T^{2} \)
17 \( 1 + (-21.6 + 12.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (39.2 + 22.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (13.9 + 79.2i)T + (-1.14e4 + 4.16e3i)T^{2} \)
29 \( 1 + (-167. + 199. i)T + (-4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-246. + 43.4i)T + (2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-70.2 - 121. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-19.4 - 23.1i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-41.0 - 112. i)T + (-6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (81.7 - 463. i)T + (-9.75e4 - 3.55e4i)T^{2} \)
53 \( 1 + 660. iT - 1.48e5T^{2} \)
59 \( 1 + (-363. - 132. i)T + (1.57e5 + 1.32e5i)T^{2} \)
61 \( 1 + (43.2 - 245. i)T + (-2.13e5 - 7.76e4i)T^{2} \)
67 \( 1 + (394. + 469. i)T + (-5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (-145. - 252. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-302. + 523. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-81.0 + 96.6i)T + (-8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-676. - 567. i)T + (9.92e4 + 5.63e5i)T^{2} \)
89 \( 1 + (5.70 + 3.29i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (634. - 230. i)T + (6.99e5 - 5.86e5i)T^{2} \)
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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

−12.77133032471638116829806623013, −11.69874188399552947539277123653, −10.98708321623054977826265369053, −10.25233983657419865995174850844, −8.279543477247487107873695589140, −7.897158882813914512444098208018, −6.39129900353778980165497006469, −4.75968650212531480229945537952, −2.59432507385150504952300692089, −0.967214092730377923128989889553, 1.17536528016810418036730413131, 4.38630273589306912199428598202, 5.39887197161209375609427289612, 6.63427879109314327198075949123, 8.178196262231903668204143378871, 8.901741735984548252217763150497, 10.36591366051696521453544435871, 10.99504697049406550450179759485, 11.98379238843860515904642090866, 13.76971873442129072888328426201

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