Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.358 - 0.933i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.88 + 8.45i)5-s + (−68.3 − 118. i)7-s + (326. + 565. i)11-s + (−125. + 216. i)13-s − 249.·17-s − 1.75e3·19-s + (−827. + 1.43e3i)23-s + (1.51e3 + 2.62e3i)25-s + (2.12e3 + 3.67e3i)29-s + (−4.49e3 + 7.78e3i)31-s + 1.33e3·35-s − 6.00e3·37-s + (−5.37e3 + 9.30e3i)41-s + (−5.02e3 − 8.70e3i)43-s + (1.17e4 + 2.03e4i)47-s + ⋯
L(s)  = 1  + (−0.0873 + 0.151i)5-s + (−0.527 − 0.912i)7-s + (0.813 + 1.40i)11-s + (−0.205 + 0.356i)13-s − 0.209·17-s − 1.11·19-s + (−0.326 + 0.564i)23-s + (0.484 + 0.839i)25-s + (0.468 + 0.812i)29-s + (−0.839 + 1.45i)31-s + 0.184·35-s − 0.720·37-s + (−0.499 + 0.864i)41-s + (−0.414 − 0.717i)43-s + (0.775 + 1.34i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.358 - 0.933i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (37, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ -0.358 - 0.933i)\)
\(L(3)\)  \(\approx\)  \(0.571807 + 0.831916i\)
\(L(\frac12)\)  \(\approx\)  \(0.571807 + 0.831916i\)
\(L(\frac{7}{2})\)   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 \( 1 \)
3 \( 1 \)
good5 \( 1 + (4.88 - 8.45i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (68.3 + 118. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-326. - 565. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (125. - 216. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 249.T + 1.41e6T^{2} \)
19 \( 1 + 1.75e3T + 2.47e6T^{2} \)
23 \( 1 + (827. - 1.43e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.12e3 - 3.67e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (4.49e3 - 7.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 6.00e3T + 6.93e7T^{2} \)
41 \( 1 + (5.37e3 - 9.30e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.02e3 + 8.70e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-1.17e4 - 2.03e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 9.41e3T + 4.18e8T^{2} \)
59 \( 1 + (-2.20e4 + 3.82e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.12e4 - 1.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.80e4 + 3.11e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.85e4T + 1.80e9T^{2} \)
73 \( 1 - 6.13e4T + 2.07e9T^{2} \)
79 \( 1 + (1.37e4 + 2.38e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (3.24e4 + 5.61e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 3.46e4T + 5.58e9T^{2} \)
97 \( 1 + (8.05e3 + 1.39e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−12.98344022465876466591283342010, −12.17170334945121295646301599198, −10.85929223596314842171622738337, −9.935471737481973468424735594929, −8.891425190804757970747453945262, −7.24360578887385279231446473004, −6.66463133888318735822297035431, −4.76258887522911509754484539289, −3.59087565741181944355999668479, −1.64418581530981571673711098479, 0.37801632988307379797940395377, 2.47599770210853296893627410271, 3.96878384367455217808690191139, 5.69041805596001822141916876797, 6.55803806683671538931144961727, 8.332486132836969501972482564456, 8.988605172340460132002030937548, 10.31632073179217040841054791435, 11.51036742326885948466366051980, 12.39424746005806104815727381912

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