Properties

Label 2-12e3-9.2-c2-0-4
Degree $2$
Conductor $1728$
Sign $-0.118 - 0.993i$
Analytic cond. $47.0845$
Root an. cond. $6.86182$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.05 + 1.18i)5-s + (4.05 − 7.02i)7-s + (−17.6 − 10.1i)11-s + (3.05 + 5.29i)13-s − 17.9i·17-s − 9.11·19-s + (−29.0 + 16.7i)23-s + (−9.67 + 16.7i)25-s + (14.4 + 8.31i)29-s + (11.1 + 19.3i)31-s + 19.2i·35-s + 50.4·37-s + (−29.9 + 17.3i)41-s + (11.5 − 19.9i)43-s + (33.1 + 19.1i)47-s + ⋯
L(s)  = 1  + (−0.411 + 0.237i)5-s + (0.579 − 1.00i)7-s + (−1.60 − 0.924i)11-s + (0.235 + 0.407i)13-s − 1.05i·17-s − 0.479·19-s + (−1.26 + 0.729i)23-s + (−0.387 + 0.670i)25-s + (0.496 + 0.286i)29-s + (0.360 + 0.624i)31-s + 0.551i·35-s + 1.36·37-s + (−0.730 + 0.421i)41-s + (0.267 − 0.463i)43-s + (0.705 + 0.407i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.118 - 0.993i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.118 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7035483925\)
\(L(\frac12)\) \(\approx\) \(0.7035483925\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (2.05 - 1.18i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.05 + 7.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (17.6 + 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.05 - 5.29i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 17.9iT - 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (29.0 - 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.4 - 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.1 - 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 50.4T + 1.36e3T^{2} \)
41 \( 1 + (29.9 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 + 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33.1 - 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (2.96 - 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (23.1 - 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.14 + 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3T + 5.32e3T^{2} \)
79 \( 1 + (-42.2 + 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (33.1 + 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (40.3 - 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.371053866061854889041237185883, −8.306945726575049008186675057425, −7.77388664668028908862068282295, −7.20456457604045536364605883655, −6.11213311510625734424354166258, −5.21208870703570201545509928836, −4.36501552705546187632614420598, −3.45060601415469129935581441389, −2.46108088740771787740585238906, −1.01125906820547450643800429045, 0.20760887160761254164935793345, 2.00147730258222819063180793169, 2.59204200890360450930315458894, 4.07121394086616238684426293341, 4.76857649865393333233312737094, 5.67215948941419450408283141530, 6.34189395484854486795895805058, 7.71240530600281150655740822215, 8.120699183275643085851911555361, 8.625914908707607682457055299819

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