Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.607 + 0.794i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 + 2.46i)2-s + (−4.17 + 6.82i)4-s + (−14.6 − 8.45i)5-s + (−3.08 + 1.78i)7-s + (−22.6 − 0.866i)8-s + (0.610 − 47.8i)10-s + (−25.0 − 43.4i)11-s + (−18.9 + 32.9i)13-s + (−8.67 − 5.15i)14-s + (−29.1 − 56.9i)16-s + 84.3i·17-s − 62.9i·19-s + (118. − 64.6i)20-s + (72.4 − 121. i)22-s + (−37.6 + 65.1i)23-s + ⋯
L(s)  = 1  + (0.488 + 0.872i)2-s + (−0.521 + 0.852i)4-s + (−1.31 − 0.756i)5-s + (−0.166 + 0.0963i)7-s + (−0.999 − 0.0382i)8-s + (0.0193 − 1.51i)10-s + (−0.687 − 1.19i)11-s + (−0.405 + 0.701i)13-s + (−0.165 − 0.0984i)14-s + (−0.455 − 0.890i)16-s + 1.20i·17-s − 0.759i·19-s + (1.32 − 0.722i)20-s + (0.702 − 1.18i)22-s + (−0.340 + 0.590i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.607 + 0.794i$
motivic weight  =  \(3\)
character  :  $\chi_{108} (71, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :3/2),\ -0.607 + 0.794i)\)
\(L(2)\)  \(\approx\)  \(0.0186925 - 0.0378506i\)
\(L(\frac12)\)  \(\approx\)  \(0.0186925 - 0.0378506i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-1.38 - 2.46i)T \)
3 \( 1 \)
good5 \( 1 + (14.6 + 8.45i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (3.08 - 1.78i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (25.0 + 43.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (18.9 - 32.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 84.3iT - 4.91e3T^{2} \)
19 \( 1 + 62.9iT - 6.85e3T^{2} \)
23 \( 1 + (37.6 - 65.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (105. - 60.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (17.2 + 9.97i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 17.7T + 5.06e4T^{2} \)
41 \( 1 + (299. + 172. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-113. + 65.3i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-153. - 265. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 479. iT - 1.48e5T^{2} \)
59 \( 1 + (-245. + 425. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (49.9 + 86.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (536. + 309. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 254.T + 3.57e5T^{2} \)
73 \( 1 - 100.T + 3.89e5T^{2} \)
79 \( 1 + (-856. + 494. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (251. + 436. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.01e3iT - 7.04e5T^{2} \)
97 \( 1 + (503. + 872. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.88900745668237929587582965293, −12.91036760309411612437794183743, −12.08356644339346729712960413695, −11.05398357845433157497185261963, −9.093459351810845127552366375561, −8.272627947224520644970902557914, −7.37273632771582259347644897560, −5.88051110030242138964180235051, −4.59854081797157130022637184566, −3.46200856258598599495476040616, 0.01913878939440817432839098584, 2.60627940952550749518246007842, 3.87146417260843868717237697899, 5.13548586166689073673740883589, 6.96200673824942370278705894599, 8.029277824829719504466927870608, 9.776522038276338312251226682650, 10.54371755693643932023070651636, 11.66480320701666160745219811041, 12.32072044908719405830797445386

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