Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $-0.849 - 0.526i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 3.78i)2-s + (−12.6 − 9.79i)4-s + (10.5 + 18.3i)5-s + (38.6 + 22.3i)7-s + (53.4 − 35.2i)8-s + (−83.0 + 16.3i)10-s + (58.6 + 33.8i)11-s + (14.5 + 25.2i)13-s + (−134. + 117. i)14-s + (64.2 + 247. i)16-s − 402.·17-s + 644. i·19-s + (45.5 − 335. i)20-s + (−204. + 178. i)22-s + (−335. + 193. i)23-s + ⋯
L(s)  = 1  + (−0.323 + 0.946i)2-s + (−0.790 − 0.611i)4-s + (0.423 + 0.732i)5-s + (0.788 + 0.455i)7-s + (0.834 − 0.550i)8-s + (−0.830 + 0.163i)10-s + (0.485 + 0.280i)11-s + (0.0861 + 0.149i)13-s + (−0.685 + 0.599i)14-s + (0.251 + 0.967i)16-s − 1.39·17-s + 1.78i·19-s + (0.113 − 0.838i)20-s + (−0.421 + 0.368i)22-s + (−0.634 + 0.366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $-0.849 - 0.526i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ -0.849 - 0.526i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.371845 + 1.30538i\)
\(L(\frac12)\)  \(\approx\)  \(0.371845 + 1.30538i\)
\(L(3)\)   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 \( 1 + (1.29 - 3.78i)T \)
3 \( 1 \)
good5 \( 1 + (-10.5 - 18.3i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 + (-38.6 - 22.3i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (-58.6 - 33.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-14.5 - 25.2i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 402.T + 8.35e4T^{2} \)
19 \( 1 - 644. iT - 1.30e5T^{2} \)
23 \( 1 + (335. - 193. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-362. + 627. i)T + (-3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (1.09e3 - 629. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 1.40e3T + 1.87e6T^{2} \)
41 \( 1 + (-774. - 1.34e3i)T + (-1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (-1.62e3 - 935. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-3.61e3 - 2.08e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 906.T + 7.89e6T^{2} \)
59 \( 1 + (-3.91e3 + 2.26e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.31e3 - 2.27e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (58.7 - 33.8i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.31e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.47e3T + 2.83e7T^{2} \)
79 \( 1 + (3.78e3 + 2.18e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-659. - 381. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 8.08e3T + 6.27e7T^{2} \)
97 \( 1 + (3.33e3 - 5.77e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−13.96559972610681525063021290496, −12.47519838740504005604567330181, −11.09388103064434837030823611345, −10.06613461980840057920001413832, −8.944806346168269796454002419042, −7.890844799906494145957632465684, −6.66188002168816444707415750801, −5.70901715568714305069996290031, −4.22027039599650065111677259523, −1.84445677763305183192815188529, 0.70287015394100756112881874485, 2.15170497034481339173308338918, 4.10050268638531902357276204450, 5.16516369785884078157061363139, 7.16482059273544113193611453938, 8.689319540926506898346632779880, 9.147703978831927017805098464101, 10.67646858624949170073547863468, 11.30219705901941350098554262818, 12.52773107196644226366868874140

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