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Properties

Label 2-728-728.347-c0-0-1
Degree $2$
Conductor $728$
Sign $-0.325 + 0.945i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s − i·7-s − 8-s + (−0.866 + 0.5i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + i·13-s + i·14-s + (−0.866 − 0.499i)15-s + 16-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)20-s + ⋯
L(s)  = 1  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (0.866 − 0.5i)5-s + (0.5 + 0.866i)6-s − i·7-s − 8-s + (−0.866 + 0.5i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + i·13-s + i·14-s + (−0.866 − 0.499i)15-s + 16-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.325 + 0.945i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ -0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5903606157\)
\(L(\frac12)\) \(\approx\) \(0.5903606157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.17507515131040933052316835257, −9.494489990792174281475797617064, −8.748832012791474453109619800538, −7.51526895009016576950254481589, −7.09425446526020399795760897624, −6.10856531479308486013993950020, −5.35381160415002680388488451279, −3.59574846213024905749184733030, −1.94948537325343994488910571721, −0.961326731060968662546534500291, 2.06261904139629663411077309125, 2.94782009069707185549142777487, 4.73451492520161941929283758894, 5.73936007860252147717166566832, 6.29144174588415031025679057351, 7.59244321616076205698098050892, 8.379842001934456829070862748877, 9.540038732425118209358276689250, 9.985327483338032861826747885603, 10.53296999966859038433973004297

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