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Properties

Label 2-12e3-9.2-c2-0-17
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

\(L(\frac{3}{2})\) \(\approx\) \(1.772295750\)
\(L(\frac12)\) \(\approx\) \(1.772295750\)
\(L(2)\) 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

−9.358741712538943927085379339831, −8.808861169727431479469490563859, −7.66486028240000518856111496832, −6.82248087128381082768735260142, −6.37820347387952298095029936244, −5.22842794149304591498578158226, −4.35727904898428918456104389303, −3.37577529330095859579972612826, −2.46600748760413630053698014627, −1.16592441018515295145176164184, 0.56191598081623322359681382188, 1.41936468412941203688483587870, 3.28781930031640066246578010008, 3.70672378996668540474195173020, 4.61062014964439474678401348619, 5.86678053727415801639265798965, 6.52646043237131888733504415596, 7.28095592814327677863778528273, 8.217007042993824738067693950991, 8.865767162962206568065290496028

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