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Properties

Label 2-12e3-36.7-c2-0-10
Degree $2$
Conductor $1728$
Sign $-0.659 - 0.751i$
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  + (4.61 + 7.99i)5-s + (5.33 + 3.07i)7-s + (3.70 + 2.13i)11-s + (−0.869 − 1.50i)13-s − 12.3·17-s + 33.9i·19-s + (−3.35 + 1.93i)23-s + (−30.1 + 52.1i)25-s + (17.8 − 30.9i)29-s + (−38.8 + 22.4i)31-s + 56.8i·35-s + 32.7·37-s + (−21.8 − 37.8i)41-s + (33.9 + 19.5i)43-s + (39.8 + 23.0i)47-s + ⋯
L(s)  = 1  + (0.923 + 1.59i)5-s + (0.761 + 0.439i)7-s + (0.336 + 0.194i)11-s + (−0.0668 − 0.115i)13-s − 0.726·17-s + 1.78i·19-s + (−0.145 + 0.0841i)23-s + (−1.20 + 2.08i)25-s + (0.615 − 1.06i)29-s + (−1.25 + 0.723i)31-s + 1.62i·35-s + 0.884·37-s + (−0.533 − 0.923i)41-s + (0.789 + 0.455i)43-s + (0.848 + 0.489i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(47.0845\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1),\ -0.659 - 0.751i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.353662899\)
\(L(\frac12)\) \(\approx\) \(2.353662899\)
\(L(2)\) 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 \( 1 + (-4.61 - 7.99i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.33 - 3.07i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.70 - 2.13i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.869 + 1.50i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.3T + 289T^{2} \)
19 \( 1 - 33.9iT - 361T^{2} \)
23 \( 1 + (3.35 - 1.93i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.8 + 30.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (38.8 - 22.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 32.7T + 1.36e3T^{2} \)
41 \( 1 + (21.8 + 37.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-33.9 - 19.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-39.8 - 23.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 46.3T + 2.80e3T^{2} \)
59 \( 1 + (23.2 - 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (23.4 - 40.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (56.9 - 32.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 96.7iT - 5.04e3T^{2} \)
73 \( 1 + 14.0T + 5.32e3T^{2} \)
79 \( 1 + (34.3 + 19.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (81.7 + 47.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 81.8T + 7.92e3T^{2} \)
97 \( 1 + (7.99 - 13.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.536672238043067363576937955493, −8.687880869278011969898431993645, −7.67924809601756479693102727869, −7.06503164540887119943317112197, −6.00820747468359196695524584080, −5.77252009897878487202128396635, −4.41202354332267157028896748646, −3.37327481660068235427967850448, −2.33805248287173968495269700077, −1.68263941884559906278501549907, 0.59942045478884677848805183935, 1.49665848722082691681094917771, 2.49232247129193097522069327284, 4.15402313716431879414843838119, 4.75772665421406635140066331584, 5.39815591122435297042897223653, 6.35613903788868471905783968471, 7.29441971509305741568503006666, 8.271605318075359227363999153509, 9.066847653753497122574724051375

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