Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.985 + 0.169i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 − 3.86i)2-s + (−13.8 + 8.03i)4-s + (5.89 + 10.2i)5-s + (−50.5 − 29.1i)7-s + (45.4 + 45.0i)8-s + (33.2 − 33.3i)10-s + (86.9 + 50.2i)11-s + (85.3 + 147. i)13-s + (−60.1 + 225. i)14-s + (126. − 222. i)16-s + 398.·17-s + 404. i·19-s + (−163. − 93.8i)20-s + (103. − 388. i)22-s + (291. − 168. i)23-s + ⋯
L(s)  = 1  + (−0.260 − 0.965i)2-s + (−0.864 + 0.502i)4-s + (0.235 + 0.408i)5-s + (−1.03 − 0.595i)7-s + (0.709 + 0.704i)8-s + (0.332 − 0.333i)10-s + (0.718 + 0.414i)11-s + (0.504 + 0.874i)13-s + (−0.306 + 1.15i)14-s + (0.495 − 0.868i)16-s + 1.37·17-s + 1.12i·19-s + (−0.409 − 0.234i)20-s + (0.213 − 0.801i)22-s + (0.550 − 0.317i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.985 + 0.169i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.985 + 0.169i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.29788 - 0.111076i\)
\(L(\frac12)\)  \(\approx\)  \(1.29788 - 0.111076i\)
\(L(3)\)   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 \( 1 + (1.04 + 3.86i)T \)
3 \( 1 \)
good5 \( 1 + (-5.89 - 10.2i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (50.5 + 29.1i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-86.9 - 50.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-85.3 - 147. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 398.T + 8.35e4T^{2} \)
19 \( 1 - 404. iT - 1.30e5T^{2} \)
23 \( 1 + (-291. + 168. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (327. - 567. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-550. + 317. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 + (-1.23e3 - 2.13e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.93e3 - 1.11e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.51e3 - 1.45e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.29e3T + 7.89e6T^{2} \)
59 \( 1 + (1.00e3 - 578. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (2.96e3 - 5.12e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.08e3 - 1.78e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 5.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.49e3T + 2.83e7T^{2} \)
79 \( 1 + (-2.78e3 - 1.61e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-7.06e3 - 4.07e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 910.T + 6.27e7T^{2} \)
97 \( 1 + (-8.80e3 + 1.52e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.72374484792423559349577274041, −11.97033267868970792945551926019, −10.67181068000475511923208737813, −9.916252024754736520540321491121, −9.021667851779969514093350094537, −7.49319429917318999197679274277, −6.19443115002667605650377519698, −4.21519421119742884222482114663, −3.11129767363934097485983053264, −1.27023917107220694855846730497, 0.78119178750198002122944017585, 3.43824791472367143985824735550, 5.31137266527009779423032038353, 6.13994168574045392962836878609, 7.40009189168743085892532067197, 8.816024193533546317551707769875, 9.374387676510743821715118752395, 10.60778262174115832831222242781, 12.27907046099112714850801098170, 13.18010734771258828504289366997

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