Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.909 - 0.414i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.36 − 2.16i)2-s + (6.63 + 14.5i)4-s + 43.0·5-s + 14.1i·7-s + (9.18 − 63.3i)8-s + (−144. − 93.1i)10-s + 209. i·11-s − 195.·13-s + (30.5 − 47.4i)14-s + (−167. + 193. i)16-s + 303.·17-s + 274. i·19-s + (285. + 626. i)20-s + (453. − 704. i)22-s − 466. i·23-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)2-s + (0.414 + 0.909i)4-s + 1.72·5-s + 0.288i·7-s + (0.143 − 0.989i)8-s + (−1.44 − 0.931i)10-s + 1.73i·11-s − 1.15·13-s + (0.155 − 0.242i)14-s + (−0.656 + 0.754i)16-s + 1.05·17-s + 0.761i·19-s + (0.714 + 1.56i)20-s + (0.936 − 1.45i)22-s − 0.881i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.909 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.909 - 0.414i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (55, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.909 - 0.414i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.44959 + 0.314739i\)
\(L(\frac12)\)  \(\approx\)  \(1.44959 + 0.314739i\)
\(L(3)\)   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 \( 1 + (3.36 + 2.16i)T \)
3 \( 1 \)
good5 \( 1 - 43.0T + 625T^{2} \)
7 \( 1 - 14.1iT - 2.40e3T^{2} \)
11 \( 1 - 209. iT - 1.46e4T^{2} \)
13 \( 1 + 195.T + 2.85e4T^{2} \)
17 \( 1 - 303.T + 8.35e4T^{2} \)
19 \( 1 - 274. iT - 1.30e5T^{2} \)
23 \( 1 + 466. iT - 2.79e5T^{2} \)
29 \( 1 - 439.T + 7.07e5T^{2} \)
31 \( 1 - 1.22e3iT - 9.23e5T^{2} \)
37 \( 1 - 624.T + 1.87e6T^{2} \)
41 \( 1 - 203.T + 2.82e6T^{2} \)
43 \( 1 - 419. iT - 3.41e6T^{2} \)
47 \( 1 - 1.99e3iT - 4.87e6T^{2} \)
53 \( 1 - 206.T + 7.89e6T^{2} \)
59 \( 1 + 2.32e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.39e3T + 1.38e7T^{2} \)
67 \( 1 + 7.34e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.54e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.24e3T + 2.83e7T^{2} \)
79 \( 1 - 6.15e3iT - 3.89e7T^{2} \)
83 \( 1 - 213. iT - 4.74e7T^{2} \)
89 \( 1 - 8.43e3T + 6.27e7T^{2} \)
97 \( 1 + 1.02e3T + 8.85e7T^{2} \)
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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

−12.51801254460561085135193059252, −12.35421470124633693914633133040, −10.40416301930289262623432753039, −9.920341364329398268589368404385, −9.170675432380164159275781729040, −7.63330240941208949337350104479, −6.45050770740207086531237036495, −4.90564827659225132555310724299, −2.61422801154241349163862214094, −1.61806399770637892483946823285, 0.925895872414073322722767301993, 2.57480643095888072848729294451, 5.36719456848448776123763115202, 6.04704170012124700585407926030, 7.35459706992388330700065689454, 8.729284449624326022791205398042, 9.677976768462982390255062523742, 10.37383015071507466701837069734, 11.56165413716846972328893151926, 13.34511150799953831732201759980

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