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Properties

Label 2-728-8.5-c1-0-44
Degree $2$
Conductor $728$
Sign $-0.0473 + 0.998i$
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  + (−0.687 − 1.23i)2-s + 3.16i·3-s + (−1.05 + 1.69i)4-s − 1.11i·5-s + (3.91 − 2.17i)6-s − 7-s + (2.82 + 0.133i)8-s − 7.03·9-s + (−1.38 + 0.768i)10-s − 4.51i·11-s + (−5.38 − 3.33i)12-s − i·13-s + (0.687 + 1.23i)14-s + 3.54·15-s + (−1.77 − 3.58i)16-s − 4.44·17-s + ⋯
L(s)  = 1  + (−0.486 − 0.873i)2-s + 1.82i·3-s + (−0.527 + 0.849i)4-s − 0.499i·5-s + (1.59 − 0.889i)6-s − 0.377·7-s + (0.998 + 0.0473i)8-s − 2.34·9-s + (−0.436 + 0.243i)10-s − 1.36i·11-s + (−1.55 − 0.963i)12-s − 0.277i·13-s + (0.183 + 0.330i)14-s + 0.914·15-s + (−0.444 − 0.895i)16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0473 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0473 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.0473 + 0.998i$
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: \((2,\ 728,\ (\ :1/2),\ -0.0473 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399198 - 0.418550i\)
\(L(\frac12)\) \(\approx\) \(0.399198 - 0.418550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.687 + 1.23i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 3.16iT - 3T^{2} \)
5 \( 1 + 1.11iT - 5T^{2} \)
11 \( 1 + 4.51iT - 11T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 6.89iT - 19T^{2} \)
23 \( 1 + 2.66T + 23T^{2} \)
29 \( 1 - 6.01iT - 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + 2.80iT - 37T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + 4.88iT - 43T^{2} \)
47 \( 1 + 6.35T + 47T^{2} \)
53 \( 1 - 6.02iT - 53T^{2} \)
59 \( 1 + 15.1iT - 59T^{2} \)
61 \( 1 + 6.44iT - 61T^{2} \)
67 \( 1 + 5.98iT - 67T^{2} \)
71 \( 1 - 0.527T + 71T^{2} \)
73 \( 1 + 7.57T + 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 + 2.98iT - 83T^{2} \)
89 \( 1 - 5.07T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−10.20573027822960570782592776964, −9.360558328490462461600699132769, −8.779207937833111812856609358745, −8.315268143461869137091202732814, −6.54212715574428303049279302341, −5.14539783269910858871594079947, −4.55983952639144084259920584926, −3.48005394069330040551332337268, −2.77441807576272029123784343488, −0.36006272947039982984368748439, 1.45418596617170237518537343692, 2.49654741513767257070033202205, 4.41674082807959429228208904860, 5.84385022622702782903346998864, 6.60781717557432566136515186012, 6.95654037602235390681871073602, 7.914872781681681035687869862727, 8.433825190891520026768714607213, 9.661353390046216070638546311375, 10.39175342949236042447055537297

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