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  :  $\chi_{108} (19, \cdot )$
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

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

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