Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.293 + 0.956i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.27 − 1.68i)2-s + (2.34 − 7.64i)4-s − 5.83i·5-s − 8.83i·7-s + (−7.52 − 21.3i)8-s + (−9.81 − 13.2i)10-s − 23.6·11-s + 54.6·13-s + (−14.8 − 20.0i)14-s + (−52.9 − 35.8i)16-s − 117. i·17-s + 109. i·19-s + (−44.6 − 13.6i)20-s + (−53.7 + 39.7i)22-s + 33.5·23-s + ⋯
L(s)  = 1  + (0.804 − 0.594i)2-s + (0.293 − 0.956i)4-s − 0.522i·5-s − 0.476i·7-s + (−0.332 − 0.943i)8-s + (−0.310 − 0.419i)10-s − 0.647·11-s + 1.16·13-s + (−0.283 − 0.383i)14-s + (−0.828 − 0.560i)16-s − 1.67i·17-s + 1.32i·19-s + (−0.499 − 0.153i)20-s + (−0.520 + 0.384i)22-s + 0.304·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.293 + 0.956i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.293 + 0.956i)\)
\(L(2)\)  \(\approx\)  \(1.38849 - 1.87808i\)
\(L(\frac12)\)  \(\approx\)  \(1.38849 - 1.87808i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-2.27 + 1.68i)T \)
3 \( 1 \)
good5 \( 1 + 5.83iT - 125T^{2} \)
7 \( 1 + 8.83iT - 343T^{2} \)
11 \( 1 + 23.6T + 1.33e3T^{2} \)
13 \( 1 - 54.6T + 2.19e3T^{2} \)
17 \( 1 + 117. iT - 4.91e3T^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 - 33.5T + 1.21e4T^{2} \)
29 \( 1 - 40.0iT - 2.43e4T^{2} \)
31 \( 1 - 292. iT - 2.97e4T^{2} \)
37 \( 1 - 283.T + 5.06e4T^{2} \)
41 \( 1 - 367. iT - 6.89e4T^{2} \)
43 \( 1 + 323. iT - 7.95e4T^{2} \)
47 \( 1 + 66.2T + 1.03e5T^{2} \)
53 \( 1 - 158. iT - 1.48e5T^{2} \)
59 \( 1 - 848.T + 2.05e5T^{2} \)
61 \( 1 + 348.T + 2.26e5T^{2} \)
67 \( 1 - 194. iT - 3.00e5T^{2} \)
71 \( 1 + 939.T + 3.57e5T^{2} \)
73 \( 1 + 473.T + 3.89e5T^{2} \)
79 \( 1 + 273. iT - 4.93e5T^{2} \)
83 \( 1 + 338.T + 5.71e5T^{2} \)
89 \( 1 + 739. iT - 7.04e5T^{2} \)
97 \( 1 + 448.T + 9.12e5T^{2} \)
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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

−12.99514272913288679916710954234, −11.97183056743307444174954735526, −10.93412073770421745609195387104, −10.00245632918760144552662456532, −8.697440519466837600675937835845, −7.11891212005718588329604185985, −5.68193634446442663199678053730, −4.55694295564074992924115542984, −3.11382846557793374289897324461, −1.11507430563042471571746425774, 2.62597603549689278839354057073, 4.08720074572229285262052802827, 5.64613600016297141701629491487, 6.55983602433222582889748345494, 7.891159996799592203527204743376, 8.899399670200224823782689950510, 10.69415794780071366396184393247, 11.49312495539503922630464484638, 12.90953769267240281970494179356, 13.37581736105869793469184925041

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