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Properties

Label 2-12e3-144.61-c1-0-9
Degree $2$
Conductor $1728$
Sign $0.737 - 0.675i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.73 + i)5-s + (0.633 + 0.366i)7-s + (−0.767 − 2.86i)11-s + (−1.63 + 6.09i)13-s + 2.26·17-s + (0.633 + 0.633i)19-s + (−1.09 + 0.633i)23-s + (8.59 + 4.96i)25-s + (2.36 − 0.633i)29-s + (3.73 + 6.46i)31-s + (2 + 2i)35-s + (1.26 − 1.26i)37-s + (2.59 − 1.5i)41-s + (−0.330 − 1.23i)43-s + (−4.83 + 8.36i)47-s + ⋯
L(s)  = 1  + (1.66 + 0.447i)5-s + (0.239 + 0.138i)7-s + (−0.231 − 0.864i)11-s + (−0.453 + 1.69i)13-s + 0.550·17-s + (0.145 + 0.145i)19-s + (−0.228 + 0.132i)23-s + (1.71 + 0.992i)25-s + (0.439 − 0.117i)29-s + (0.670 + 1.16i)31-s + (0.338 + 0.338i)35-s + (0.208 − 0.208i)37-s + (0.405 − 0.234i)41-s + (−0.0503 − 0.187i)43-s + (−0.704 + 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.395872782\)
\(L(\frac12)\) \(\approx\) \(2.395872782\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.73 - i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.767 + 2.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.63 - 6.09i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 2.26T + 17T^{2} \)
19 \( 1 + (-0.633 - 0.633i)T + 19iT^{2} \)
23 \( 1 + (1.09 - 0.633i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.36 + 0.633i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-3.73 - 6.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.330 + 1.23i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.83 - 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.535 + 0.535i)T - 53iT^{2} \)
59 \( 1 + (-4.96 - 1.33i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3 - 0.803i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.40 - 5.23i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.36 - 0.366i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.396495848692853556889067253765, −8.879925867108037352525626572197, −7.84108644667057438093067460220, −6.75542746563733836533270037720, −6.26058785291757868369048916566, −5.45051269111685758408101942678, −4.63684809192438005860293216043, −3.26830725285268289836609606098, −2.31410539464191925640222046093, −1.44186056819739029083951029326, 0.974007487622345031718137589208, 2.15028089707263274171209692966, 2.95694359390095642655836284027, 4.49574394085364304428198265458, 5.29149361343509990926232856118, 5.79299651406834563412061220869, 6.74392525762577163173916360405, 7.76432230530868099827109902417, 8.391849645829663066942208400418, 9.512807390034400946654710434131

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