Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} $
Sign $0.508 - 0.861i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.61 + 1.70i)2-s + (10.1 − 12.3i)4-s + (2.83 − 4.90i)5-s + (−45.1 + 26.0i)7-s + (−15.6 + 62.0i)8-s + (−1.85 + 22.5i)10-s + (92.3 − 53.3i)11-s + (61.0 − 105. i)13-s + (118. − 171. i)14-s + (−49.5 − 251. i)16-s + 122.·17-s + 593. i·19-s + (−31.8 − 84.8i)20-s + (−242. + 350. i)22-s + (473. + 273. i)23-s + ⋯
L(s)  = 1  + (−0.904 + 0.427i)2-s + (0.634 − 0.772i)4-s + (0.113 − 0.196i)5-s + (−0.921 + 0.532i)7-s + (−0.244 + 0.969i)8-s + (−0.0185 + 0.225i)10-s + (0.763 − 0.440i)11-s + (0.361 − 0.625i)13-s + (0.605 − 0.874i)14-s + (−0.193 − 0.981i)16-s + 0.424·17-s + 1.64i·19-s + (−0.0796 − 0.212i)20-s + (−0.501 + 0.724i)22-s + (0.895 + 0.516i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(108\)    =    \(2^{2} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.508 - 0.861i$
motivic weight  =  \(4\)
character  :  $\chi_{108} (91, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 108,\ (\ :2),\ 0.508 - 0.861i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.906533 + 0.517486i\)
\(L(\frac12)\)  \(\approx\)  \(0.906533 + 0.517486i\)
\(L(3)\)   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 \( 1 + (3.61 - 1.70i)T \)
3 \( 1 \)
good5 \( 1 + (-2.83 + 4.90i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (45.1 - 26.0i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-92.3 + 53.3i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-61.0 + 105. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 - 122.T + 8.35e4T^{2} \)
19 \( 1 - 593. iT - 1.30e5T^{2} \)
23 \( 1 + (-473. - 273. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-367. - 637. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (507. + 292. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + (1.43e3 - 2.48e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.94e3 + 1.12e3i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (913. - 527. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 4.75e3T + 7.89e6T^{2} \)
59 \( 1 + (1.86e3 + 1.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-33.1 - 57.4i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (3.55e3 + 2.05e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 + (-1.19e3 + 690. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.60e3 + 1.50e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + (-2.40e3 - 4.16e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−13.10807529804070251876393810702, −12.00535049252811668833082006969, −10.82186166333891600505146089241, −9.688232687768417232950213390196, −8.935775866740767096680276248900, −7.77655575682107198644305689316, −6.39739025999097521274756494357, −5.55912426324291342451980842717, −3.22181921741162671277183338828, −1.19045631746223363094626627644, 0.76166590474402422530030790468, 2.66960433231473273589000474454, 4.13713111554837133701954710466, 6.51531096675121698974037482262, 7.15564462096576989159206703329, 8.777047873406504894935781193040, 9.576002331467153599881974572261, 10.57539937180594399341064195573, 11.57618357974080711384631503399, 12.65989399011706538757169809083

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