Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.974 + 0.225i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.01 + 5.28i)2-s + (−23.8 + 21.2i)4-s + (78.5 − 45.3i)5-s + (−69.6 − 40.2i)7-s + (−160. − 83.4i)8-s + (397. + 323. i)10-s + (275. − 477. i)11-s + (−307. − 532. i)13-s + (72.4 − 449. i)14-s + (118. − 1.01e3i)16-s − 482. i·17-s − 453. i·19-s + (−911. + 2.75e3i)20-s + (3.08e3 + 496. i)22-s + (310. + 538. i)23-s + ⋯
L(s)  = 1  + (0.355 + 0.934i)2-s + (−0.746 + 0.665i)4-s + (1.40 − 0.811i)5-s + (−0.537 − 0.310i)7-s + (−0.887 − 0.461i)8-s + (1.25 + 1.02i)10-s + (0.687 − 1.19i)11-s + (−0.504 − 0.873i)13-s + (0.0987 − 0.612i)14-s + (0.115 − 0.993i)16-s − 0.404i·17-s − 0.288i·19-s + (−0.509 + 1.54i)20-s + (1.35 + 0.218i)22-s + (0.122 + 0.212i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.974 + 0.225i$
motivic weight  =  \(5\)
character  :  $\chi_{108} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :5/2),\ 0.974 + 0.225i)\)
\(L(3)\)  \(\approx\)  \(2.22820 - 0.254352i\)
\(L(\frac12)\)  \(\approx\)  \(2.22820 - 0.254352i\)
\(L(\frac{7}{2})\)   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 \( 1 + (-2.01 - 5.28i)T \)
3 \( 1 \)
good5 \( 1 + (-78.5 + 45.3i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (69.6 + 40.2i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-275. + 477. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (307. + 532. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 482. iT - 1.41e6T^{2} \)
19 \( 1 + 453. iT - 2.47e6T^{2} \)
23 \( 1 + (-310. - 538. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-7.77e3 - 4.49e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (650. - 375. i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 4.22e3T + 6.93e7T^{2} \)
41 \( 1 + (1.10e4 - 6.35e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.39e4 - 8.05e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-9.57e3 + 1.65e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 8.42e3iT - 4.18e8T^{2} \)
59 \( 1 + (2.69e3 + 4.66e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (5.91e3 - 1.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.71e4 + 1.56e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.88e4T + 1.80e9T^{2} \)
73 \( 1 + 2.38e4T + 2.07e9T^{2} \)
79 \( 1 + (1.93e4 + 1.11e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.93e4 + 6.81e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 8.79e4iT - 5.58e9T^{2} \)
97 \( 1 + (3.91e4 - 6.78e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−13.13650566316408806798546303982, −12.10697853405179783507508760007, −10.26402550520285004546434370140, −9.247887779894439552336321525959, −8.442089545627034882381259446835, −6.82139767034840013071393006863, −5.84883939895834708326231357308, −4.91889834006200741389057315623, −3.13789414916495862120771134330, −0.78519996024841516153528184938, 1.73650294942705089881928292296, 2.69413860672873030490291050223, 4.39193880125759021846626939438, 5.91018100790428593371505302107, 6.80960170133987572440007761762, 9.072739167361198959757789473509, 9.825694254172456034602916598975, 10.48171378855823529953353961986, 11.90863947857376739454001472834, 12.67938521410415342235136223877

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