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Properties

Label 2-728-8.5-c1-0-50
Degree $2$
Conductor $728$
Sign $-0.821 + 0.569i$
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  + (−1.34 − 0.446i)2-s − 2.10i·3-s + (1.60 + 1.19i)4-s + 0.418i·5-s + (−0.938 + 2.81i)6-s + 7-s + (−1.61 − 2.32i)8-s − 1.41·9-s + (0.187 − 0.561i)10-s − 0.137i·11-s + (2.51 − 3.36i)12-s − i·13-s + (−1.34 − 0.446i)14-s + 0.879·15-s + (1.12 + 3.83i)16-s − 6.25·17-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)2-s − 1.21i·3-s + (0.800 + 0.599i)4-s + 0.187i·5-s + (−0.383 + 1.15i)6-s + 0.377·7-s + (−0.569 − 0.821i)8-s − 0.470·9-s + (0.0591 − 0.177i)10-s − 0.0415i·11-s + (0.727 − 0.970i)12-s − 0.277i·13-s + (−0.358 − 0.119i)14-s + 0.227·15-s + (0.280 + 0.959i)16-s − 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.821 + 0.569i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.821 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255166 - 0.815853i\)
\(L(\frac12)\) \(\approx\) \(0.255166 - 0.815853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.446i)T \)
7 \( 1 - T \)
13 \( 1 + iT \)
good3 \( 1 + 2.10iT - 3T^{2} \)
5 \( 1 - 0.418iT - 5T^{2} \)
11 \( 1 + 0.137iT - 11T^{2} \)
17 \( 1 + 6.25T + 17T^{2} \)
19 \( 1 + 4.70iT - 19T^{2} \)
23 \( 1 - 3.85T + 23T^{2} \)
29 \( 1 + 8.75iT - 29T^{2} \)
31 \( 1 + 0.493T + 31T^{2} \)
37 \( 1 + 3.24iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 4.67iT - 43T^{2} \)
47 \( 1 - 8.38T + 47T^{2} \)
53 \( 1 + 4.31iT - 53T^{2} \)
59 \( 1 - 11.5iT - 59T^{2} \)
61 \( 1 - 4.80iT - 61T^{2} \)
67 \( 1 + 2.86iT - 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 - 3.44T + 79T^{2} \)
83 \( 1 - 5.29iT - 83T^{2} \)
89 \( 1 + 6.80T + 89T^{2} \)
97 \( 1 - 5.29T + 97T^{2} \)
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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

−10.11375158945689017888124398565, −8.890520925109126675431025667002, −8.478320662056516385804910860157, −7.23411160540348331804956551405, −7.04788827175163051908192929156, −6.00454291553723017964477921088, −4.43759324661922090133086309646, −2.81422762969585004009548374798, −1.96093526764085400362748341092, −0.60805307198913000519759230178, 1.61717247501762645345673488341, 3.19077671086222038702624557055, 4.55912098669622844367318858393, 5.24031405832208135516223725323, 6.49927429956126267939398844914, 7.32527036704899549110081820675, 8.635767181584400054813890559696, 8.897031157639801983788953660996, 9.850706009681467950895685024065, 10.63102300661842316266272029706

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