Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.832 + 0.553i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.58 + 3.04i)2-s + (−2.59 − 15.7i)4-s + (5.51 − 9.55i)5-s + (10.3 − 5.95i)7-s + (54.8 + 32.9i)8-s + (14.8 + 41.5i)10-s + (−189. + 109. i)11-s + (18.5 − 32.1i)13-s + (−8.54 + 46.8i)14-s + (−242. + 82.0i)16-s − 284.·17-s − 45.4i·19-s + (−165. − 62.2i)20-s + (157. − 863. i)22-s + (−174. − 100. i)23-s + ⋯
L(s)  = 1  + (−0.647 + 0.762i)2-s + (−0.162 − 0.986i)4-s + (0.220 − 0.382i)5-s + (0.210 − 0.121i)7-s + (0.857 + 0.514i)8-s + (0.148 + 0.415i)10-s + (−1.57 + 0.906i)11-s + (0.109 − 0.189i)13-s + (−0.0435 + 0.239i)14-s + (−0.947 + 0.320i)16-s − 0.982·17-s − 0.126i·19-s + (−0.412 − 0.155i)20-s + (0.325 − 1.78i)22-s + (−0.329 − 0.190i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.832 + 0.553i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.832 + 0.553i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.00275369 - 0.00911195i\)
\(L(\frac12)\)  \(\approx\)  \(0.00275369 - 0.00911195i\)
\(L(3)\)   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 \( 1 + (2.58 - 3.04i)T \)
3 \( 1 \)
good5 \( 1 + (-5.51 + 9.55i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-10.3 + 5.95i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (189. - 109. i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-18.5 + 32.1i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 284.T + 8.35e4T^{2} \)
19 \( 1 + 45.4iT - 1.30e5T^{2} \)
23 \( 1 + (174. + 100. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (614. + 1.06e3i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (1.31e3 + 757. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + (1.31e3 - 2.28e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (34.6 - 19.9i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-2.49e3 + 1.44e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + 1.41e3T + 7.89e6T^{2} \)
59 \( 1 + (2.45e3 + 1.41e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-2.62e3 - 4.55e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-805. - 465. i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.16e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.16e3T + 2.83e7T^{2} \)
79 \( 1 + (-6.48e3 + 3.74e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-966. + 558. i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 - 6.73e3T + 6.27e7T^{2} \)
97 \( 1 + (6.02e3 + 1.04e4i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−12.88348231721127559053176468993, −11.18126422316409246434307857033, −10.22454887407854756558965056727, −9.225992754996783844050855078291, −8.060390976018622801965648698968, −7.16886781745421674897470920360, −5.67925585228160322109389852031, −4.65636480579066515975790976711, −2.02278247010574727168099804017, −0.00487938322354107417964924550, 2.08542621183132975498608919017, 3.41464776348280140312454213426, 5.23436527619831586681564984472, 6.98236437335693020633951613665, 8.218836890137430672869365649014, 9.109170114192163680208143460916, 10.65031990218839899093086188025, 10.83641285792725712263342450452, 12.27705894787072707905712779606, 13.24042367163031214515469467573

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