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Properties

Label 2-728-8.5-c1-0-58
Degree $2$
Conductor $728$
Sign $-0.623 + 0.781i$
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  + (0.961 − 1.03i)2-s − 0.167i·3-s + (−0.149 − 1.99i)4-s + 0.700i·5-s + (−0.174 − 0.161i)6-s − 7-s + (−2.21 − 1.76i)8-s + 2.97·9-s + (0.726 + 0.674i)10-s − 5.99i·11-s + (−0.334 + 0.0251i)12-s + i·13-s + (−0.961 + 1.03i)14-s + 0.117·15-s + (−3.95 + 0.596i)16-s − 6.07·17-s + ⋯
L(s)  = 1  + (0.680 − 0.733i)2-s − 0.0969i·3-s + (−0.0747 − 0.997i)4-s + 0.313i·5-s + (−0.0710 − 0.0659i)6-s − 0.377·7-s + (−0.781 − 0.623i)8-s + 0.990·9-s + (0.229 + 0.213i)10-s − 1.80i·11-s + (−0.0966 + 0.00724i)12-s + 0.277i·13-s + (−0.257 + 0.277i)14-s + 0.0303·15-s + (−0.988 + 0.149i)16-s − 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.623 + 0.781i$
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: \((2,\ 728,\ (\ :1/2),\ -0.623 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844783 - 1.75407i\)
\(L(\frac12)\) \(\approx\) \(0.844783 - 1.75407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.961 + 1.03i)T \)
7 \( 1 + T \)
13 \( 1 - iT \)
good3 \( 1 + 0.167iT - 3T^{2} \)
5 \( 1 - 0.700iT - 5T^{2} \)
11 \( 1 + 5.99iT - 11T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 + 5.38iT - 19T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
29 \( 1 - 1.72iT - 29T^{2} \)
31 \( 1 + 0.961T + 31T^{2} \)
37 \( 1 + 9.96iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 - 1.29iT - 43T^{2} \)
47 \( 1 - 7.75T + 47T^{2} \)
53 \( 1 - 4.96iT - 53T^{2} \)
59 \( 1 - 0.526iT - 59T^{2} \)
61 \( 1 - 9.77iT - 61T^{2} \)
67 \( 1 - 10.9iT - 67T^{2} \)
71 \( 1 - 2.78T + 71T^{2} \)
73 \( 1 - 8.46T + 73T^{2} \)
79 \( 1 - 2.94T + 79T^{2} \)
83 \( 1 - 5.06iT - 83T^{2} \)
89 \( 1 - 6.28T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.54240514251543731579418903524, −9.126117434295871822910170062080, −8.858595478067145037037441925290, −7.02810331965402196788005767794, −6.60143019706405223192976281677, −5.44456859933656216390998565068, −4.44123737964252316602556061993, −3.40956229466204613661299117196, −2.46933270312047696079794664492, −0.830087227141880809343139296873, 1.98959856827154455710165565898, 3.49507903960624433374664006428, 4.60807122521669917689991052658, 5.00009766018804031166941729288, 6.56598206300232719856665372927, 6.94301006353463034789256824598, 7.898190446203214302492480047606, 8.924848237593223760347406593634, 9.741724997643531309497736379976, 10.60204263549623807969410926385

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