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Properties

Label 2-728-728.69-c0-0-1
Degree $2$
Conductor $728$
Sign $0.246 - 0.969i$
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 + (−0.258 + 0.448i)3-s + (0.499 + 0.866i)4-s + 1.93i·5-s + (0.448 − 0.258i)6-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (0.366 + 0.633i)9-s + (0.965 − 1.67i)10-s − 0.517·12-s + (−0.965 − 0.258i)13-s − 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s − 0.732i·18-s + (1.22 − 0.707i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.258 + 0.448i)3-s + (0.499 + 0.866i)4-s + 1.93i·5-s + (0.448 − 0.258i)6-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (0.366 + 0.633i)9-s + (0.965 − 1.67i)10-s − 0.517·12-s + (−0.965 − 0.258i)13-s − 0.999·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s − 0.732i·18-s + (1.22 − 0.707i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6316054555\)
\(L(\frac12)\) \(\approx\) \(0.6316054555\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
good3 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - 1.93iT - T^{2} \)
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 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.58543308202511179078883170484, −10.17457436180487690047781807134, −9.504661962708371551873735452683, −7.937521972672067570120090972589, −7.43677946672101651243688122828, −6.86738577435185021143988158593, −5.40156154980029910705527212029, −4.08322555415952045209187290288, −3.02634872185218437901943661811, −2.00846403210482872224541483087, 0.999335107853771514727881998816, 1.97902552396715986832103889776, 4.37701598004499685478259843175, 5.25453226437982872632200836040, 5.88830971471864624067531878422, 7.18566618017711682929437395111, 7.991689717032934427660304361021, 8.631628855081032411188543433049, 9.426829181694240812676430130186, 9.990567912522789539437629823017

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