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Properties

Label 2-728-13.4-c1-0-12
Degree $2$
Conductor $728$
Sign $0.956 + 0.290i$
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  + (1.01 − 1.76i)3-s + 2.24i·5-s + (−0.866 + 0.5i)7-s + (−0.570 − 0.987i)9-s + (0.952 + 0.549i)11-s + (1.09 − 3.43i)13-s + (3.95 + 2.28i)15-s + (2.59 + 4.49i)17-s + (5.17 − 2.98i)19-s + 2.03i·21-s + (4.03 − 6.98i)23-s − 0.0486·25-s + 3.78·27-s + (−2.88 + 4.99i)29-s + 2.86i·31-s + ⋯
L(s)  = 1  + (0.587 − 1.01i)3-s + 1.00i·5-s + (−0.327 + 0.188i)7-s + (−0.190 − 0.329i)9-s + (0.287 + 0.165i)11-s + (0.302 − 0.953i)13-s + (1.02 + 0.590i)15-s + (0.629 + 1.08i)17-s + (1.18 − 0.685i)19-s + 0.444i·21-s + (0.841 − 1.45i)23-s − 0.00972·25-s + 0.728·27-s + (−0.535 + 0.927i)29-s + 0.515i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.956 + 0.290i$
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: \((2,\ 728,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91290 - 0.283959i\)
\(L(\frac12)\) \(\approx\) \(1.91290 - 0.283959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 + 3.43i)T \)
good3 \( 1 + (-1.01 + 1.76i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.24iT - 5T^{2} \)
11 \( 1 + (-0.952 - 0.549i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.59 - 4.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.17 + 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.03 + 6.98i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.88 - 4.99i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.86iT - 31T^{2} \)
37 \( 1 + (-9.73 - 5.61i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.08 + 5.24i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.38 + 4.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + (-4.24 + 2.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.06 + 8.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.00 + 1.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.77 + 1.60i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.50iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 3.92iT - 83T^{2} \)
89 \( 1 + (-9.61 - 5.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.46 - 4.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.50084312840747019832580343508, −9.443961607176938286542706798688, −8.416270046176458697820247615298, −7.72127735163707016523307551089, −6.86090454688828023036178163836, −6.30377544837512972017651910533, −5.00874676259589311826453421107, −3.29729262022380101036769448447, −2.78998735891384042244113165224, −1.32097907765020360070341530925, 1.24054272776326575396413256480, 3.11399416867314768165161121159, 3.93588110910916582410981248492, 4.83938547357438973525961535143, 5.72872813238027787269965910948, 7.07000221682069427666250579492, 8.031910610474940199239639656905, 9.081093374230954936067899233107, 9.488431092091467177447521798124, 9.988550671812884454509970779486

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