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Properties

Label 2-728-8.5-c1-0-47
Degree $2$
Conductor $728$
Sign $0.924 + 0.382i$
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  + (1.30 − 0.541i)2-s + 1.08i·3-s + (1.41 − 1.41i)4-s − 0.821i·5-s + (0.589 + 1.42i)6-s − 7-s + (1.08 − 2.61i)8-s + 1.81·9-s + (−0.444 − 1.07i)10-s + 4.25i·11-s + (1.54 + 1.53i)12-s − i·13-s + (−1.30 + 0.541i)14-s + 0.894·15-s + (−0.00272 − 3.99i)16-s + 7.72·17-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + 0.628i·3-s + (0.706 − 0.707i)4-s − 0.367i·5-s + (0.240 + 0.580i)6-s − 0.377·7-s + (0.382 − 0.924i)8-s + 0.604·9-s + (−0.140 − 0.339i)10-s + 1.28i·11-s + (0.444 + 0.444i)12-s − 0.277i·13-s + (−0.349 + 0.144i)14-s + 0.230·15-s + (−0.000682 − 0.999i)16-s + 1.87·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.924 + 0.382i$
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: \((2,\ 728,\ (\ :1/2),\ 0.924 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74805 - 0.545892i\)
\(L(\frac12)\) \(\approx\) \(2.74805 - 0.545892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.541i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 1.08iT - 3T^{2} \)
5 \( 1 + 0.821iT - 5T^{2} \)
11 \( 1 - 4.25iT - 11T^{2} \)
17 \( 1 - 7.72T + 17T^{2} \)
19 \( 1 + 6.99iT - 19T^{2} \)
23 \( 1 - 1.19T + 23T^{2} \)
29 \( 1 - 4.38iT - 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 + 1.17iT - 37T^{2} \)
41 \( 1 + 7.88T + 41T^{2} \)
43 \( 1 + 4.44iT - 43T^{2} \)
47 \( 1 + 7.93T + 47T^{2} \)
53 \( 1 - 2.16iT - 53T^{2} \)
59 \( 1 - 2.66iT - 59T^{2} \)
61 \( 1 - 4.10iT - 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 5.93T + 73T^{2} \)
79 \( 1 + 9.85T + 79T^{2} \)
83 \( 1 + 0.116iT - 83T^{2} \)
89 \( 1 + 7.88T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−10.26679322363327986644331318831, −9.820726169691805404185628227326, −8.955267522863131981035106125953, −7.33571368794075502610900915809, −6.89167118272583046210880374936, −5.36226447044484395812445933684, −4.92322161813646082158733468045, −3.89581850213422494424418177312, −2.93291984143925848997069350077, −1.39725593137716549449608020454, 1.55787893248016376484597352790, 3.14362375156268040611170028359, 3.75825537710333854320186694108, 5.24867092518772475543537999142, 6.08347401601862449899772213838, 6.75574262634541141784447577364, 7.74516980800171270468780041408, 8.247764264831616296711021654153, 9.714331884724367881585799613384, 10.59145236070773369090660348529

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