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Properties

Label 2-728-13.4-c1-0-1
Degree $2$
Conductor $728$
Sign $-0.751 - 0.659i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
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  + (−0.509 + 0.883i)3-s − 2.02i·5-s + (−0.866 + 0.5i)7-s + (0.979 + 1.69i)9-s + (−5.06 − 2.92i)11-s + (−2.41 + 2.68i)13-s + (1.78 + 1.03i)15-s + (3.04 + 5.26i)17-s + (−0.610 + 0.352i)19-s − 1.01i·21-s + (−2.48 + 4.30i)23-s + 0.919·25-s − 5.05·27-s + (−4.92 + 8.53i)29-s − 4.81i·31-s + ⋯
L(s)  = 1  + (−0.294 + 0.509i)3-s − 0.903i·5-s + (−0.327 + 0.188i)7-s + (0.326 + 0.565i)9-s + (−1.52 − 0.881i)11-s + (−0.668 + 0.743i)13-s + (0.460 + 0.265i)15-s + (0.737 + 1.27i)17-s + (−0.139 + 0.0808i)19-s − 0.222i·21-s + (−0.517 + 0.897i)23-s + 0.183·25-s − 0.973·27-s + (−0.914 + 1.58i)29-s − 0.864i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200103 + 0.531798i\)
\(L(\frac12)\) \(\approx\) \(0.200103 + 0.531798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (2.41 - 2.68i)T \)
good3 \( 1 + (0.509 - 0.883i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.02iT - 5T^{2} \)
11 \( 1 + (5.06 + 2.92i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.04 - 5.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.610 - 0.352i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.48 - 4.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.92 - 8.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.81iT - 31T^{2} \)
37 \( 1 + (-1.45 - 0.838i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.89 + 1.09i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.391 - 0.677i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.26iT - 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + (5.03 - 2.90i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.48 + 6.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.7 - 6.22i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 - 1.05T + 79T^{2} \)
83 \( 1 + 3.04iT - 83T^{2} \)
89 \( 1 + (1.34 + 0.775i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.33 + 0.767i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.65672006636119345169524707125, −9.924737762611939785505765879979, −9.094842683458798753708430707645, −8.131032156274490636291435827059, −7.48953060698649360396748519462, −5.95692814710090517617079326762, −5.31730820815345411938065788699, −4.51074581639131382542107280841, −3.30437609375132688338027008223, −1.77732907451972688249686981972, 0.28757343407275822317073358684, 2.33740334738522656758597591762, 3.19619734445304582112153965324, 4.67218786923450212878991788166, 5.67716397997850005968009950442, 6.72485270705195221624971040925, 7.38363845827066758305439172868, 7.947992014248597005626894001580, 9.560114367038726217958559007594, 10.07612984570341574322782992655

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