Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.733 - 0.679i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.53 − 7.09i)3-s + (−14.0 − 38.6i)5-s + (−5.57 − 31.6i)7-s + (−19.7 + 78.5i)9-s + (−4.11 + 11.3i)11-s + (91.3 + 76.6i)13-s + (−196. + 313. i)15-s + (−175. − 101. i)17-s + (98.3 + 170. i)19-s + (−193. + 214. i)21-s + (−707. − 124. i)23-s + (−814. + 683. i)25-s + (666. − 294. i)27-s + (523. + 623. i)29-s + (28.4 − 161. i)31-s + ⋯
L(s)  = 1  + (−0.614 − 0.788i)3-s + (−0.562 − 1.54i)5-s + (−0.113 − 0.645i)7-s + (−0.243 + 0.969i)9-s + (−0.0340 + 0.0934i)11-s + (0.540 + 0.453i)13-s + (−0.872 + 1.39i)15-s + (−0.606 − 0.350i)17-s + (0.272 + 0.471i)19-s + (−0.438 + 0.486i)21-s + (−1.33 − 0.235i)23-s + (−1.30 + 1.09i)25-s + (0.914 − 0.404i)27-s + (0.621 + 0.741i)29-s + (0.0295 − 0.167i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.733 - 0.679i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.733 - 0.679i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.165487 + 0.422151i\)
\(L(\frac12)\)  \(\approx\)  \(0.165487 + 0.422151i\)
\(L(3)\)   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 \( 1 + (5.53 + 7.09i)T \)
good5 \( 1 + (14.0 + 38.6i)T + (-478. + 401. i)T^{2} \)
7 \( 1 + (5.57 + 31.6i)T + (-2.25e3 + 821. i)T^{2} \)
11 \( 1 + (4.11 - 11.3i)T + (-1.12e4 - 9.41e3i)T^{2} \)
13 \( 1 + (-91.3 - 76.6i)T + (4.95e3 + 2.81e4i)T^{2} \)
17 \( 1 + (175. + 101. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-98.3 - 170. i)T + (-6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (707. + 124. i)T + (2.62e5 + 9.57e4i)T^{2} \)
29 \( 1 + (-523. - 623. i)T + (-1.22e5 + 6.96e5i)T^{2} \)
31 \( 1 + (-28.4 + 161. i)T + (-8.67e5 - 3.15e5i)T^{2} \)
37 \( 1 + (941. - 1.63e3i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + (-1.75e3 + 2.09e3i)T + (-4.90e5 - 2.78e6i)T^{2} \)
43 \( 1 + (3.24e3 + 1.18e3i)T + (2.61e6 + 2.19e6i)T^{2} \)
47 \( 1 + (-89.3 + 15.7i)T + (4.58e6 - 1.66e6i)T^{2} \)
53 \( 1 + 844. iT - 7.89e6T^{2} \)
59 \( 1 + (1.83e3 + 5.03e3i)T + (-9.28e6 + 7.78e6i)T^{2} \)
61 \( 1 + (-860. - 4.87e3i)T + (-1.30e7 + 4.73e6i)T^{2} \)
67 \( 1 + (-561. - 471. i)T + (3.49e6 + 1.98e7i)T^{2} \)
71 \( 1 + (3.35e3 + 1.93e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-1.19e3 - 2.07e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-3.23e3 + 2.71e3i)T + (6.76e6 - 3.83e7i)T^{2} \)
83 \( 1 + (6.78e3 + 8.08e3i)T + (-8.24e6 + 4.67e7i)T^{2} \)
89 \( 1 + (-4.41e3 + 2.54e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + (1.46e4 + 5.33e3i)T + (6.78e7 + 5.69e7i)T^{2} \)
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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

−12.26244638288857907367204134403, −11.67025642732434657494904138539, −10.33467806964381252909477749043, −8.799438883588181928916051507357, −7.926426746094532964331715688325, −6.70110124559067967694154390062, −5.27980405859935633875904824017, −4.14314805738066368884866053030, −1.51287988225266472831072352745, −0.22418818057956879627212487659, 2.86681213948107983196900358530, 4.06845772702838920231824939461, 5.80031356002396716692224026895, 6.69572857354088437686154575763, 8.181785149240534933768959563028, 9.633988206044259962749567286166, 10.66209756303773333607960610623, 11.31642269857606763975630958038, 12.23409647761603911851074097330, 13.82944786944331328392549763965

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