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Properties

Label 2-728-728.517-c0-0-3
Degree $2$
Conductor $728$
Sign $-0.969 + 0.246i$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.965 − 1.67i)3-s + (0.499 − 0.866i)4-s + 0.517i·5-s + (−1.67 − 0.965i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−1.36 + 2.36i)9-s + (0.258 + 0.448i)10-s − 1.93·12-s + (−0.258 − 0.965i)13-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 2.73i·18-s + (1.22 + 0.707i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.965 − 1.67i)3-s + (0.499 − 0.866i)4-s + 0.517i·5-s + (−1.67 − 0.965i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−1.36 + 2.36i)9-s + (0.258 + 0.448i)10-s − 1.93·12-s + (−0.258 − 0.965i)13-s − 0.999·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + 2.73i·18-s + (1.22 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ -0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9727353252\)
\(L(\frac12)\) \(\approx\) \(0.9727353252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.258 + 0.965i)T \)
good3 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 0.517iT - T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−10.58784810762524686326876202046, −9.822387998177571423636545788583, −8.039649174695979859371199647945, −7.19549683533480476332749574981, −6.57542422545948173032171684454, −5.86914294820299296504722163499, −5.03448634276592809840388836677, −3.37298153125330783713215686489, −2.39438837219290775156278920171, −0.912711557107462256778664155386, 2.99317266879906967942957282813, 3.91153913587727062374076781707, 4.79794109337177889767726832972, 5.44708159964282805717245528768, 6.20537830094755422218196439131, 7.14149840591360225423978517126, 8.769495308952641826636423203483, 9.312494585354325145330862119746, 10.09590946888574478561337966873, 11.23647710904880353326443204830

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