Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.933 + 0.358i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.05 − 2.57i)2-s + (2.72 − 15.7i)4-s + (−14.3 − 24.7i)5-s + (22.2 + 12.8i)7-s + (−32.3 − 55.2i)8-s + (−107. − 38.9i)10-s + (−93.9 − 54.2i)11-s + (44.2 + 76.5i)13-s + (101. − 17.9i)14-s + (−241. − 85.8i)16-s − 504.·17-s − 191. i·19-s + (−429. + 158. i)20-s + (−427. + 76.1i)22-s + (831. − 480. i)23-s + ⋯
L(s)  = 1  + (0.764 − 0.644i)2-s + (0.170 − 0.985i)4-s + (−0.572 − 0.991i)5-s + (0.453 + 0.261i)7-s + (−0.504 − 0.863i)8-s + (−1.07 − 0.389i)10-s + (−0.776 − 0.448i)11-s + (0.261 + 0.453i)13-s + (0.515 − 0.0918i)14-s + (−0.942 − 0.335i)16-s − 1.74·17-s − 0.530i·19-s + (−1.07 + 0.395i)20-s + (−0.882 + 0.157i)22-s + (1.57 − 0.907i)23-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.933 + 0.358i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.933 + 0.358i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.348227 - 1.87938i\)
\(L(\frac12)\)  \(\approx\)  \(0.348227 - 1.87938i\)
\(L(3)\)   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 \( 1 + (-3.05 + 2.57i)T \)
3 \( 1 \)
good5 \( 1 + (14.3 + 24.7i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-22.2 - 12.8i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (93.9 + 54.2i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-44.2 - 76.5i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 504.T + 8.35e4T^{2} \)
19 \( 1 + 191. iT - 1.30e5T^{2} \)
23 \( 1 + (-831. + 480. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-396. + 687. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (285. - 164. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 - 209.T + 1.87e6T^{2} \)
41 \( 1 + (-528. - 914. i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-2.88e3 - 1.66e3i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-977. - 564. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 1.13e3T + 7.89e6T^{2} \)
59 \( 1 + (-4.03e3 + 2.33e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.79e3 + 4.84e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-6.12e3 + 3.53e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 + (1.52e3 + 879. i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (2.62e3 + 1.51e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 559.T + 6.27e7T^{2} \)
97 \( 1 + (-1.10e3 + 1.90e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−12.72574874684154893569735902433, −11.42346481220820113230188175165, −10.92281455808950614095220855338, −9.243806598389651254007357554890, −8.363325964487692113425673874478, −6.60788093935198423327043695747, −5.06868609070256536020978165305, −4.32610788062708071370311770569, −2.49279553383707931645145450583, −0.65859066781221928549747609964, 2.68809270268556389292393620842, 4.04895547609738533655711272473, 5.37065342996470640757533284895, 6.90697896588268558947198241150, 7.52883291767212073974997069600, 8.789025963538064972841674084327, 10.72839806190527451696355288332, 11.28853940207651931277128480272, 12.66416167122806011578200541622, 13.52422124295123544839435121662

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