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Properties

Label 2-728-728.347-c0-0-1
Degree 22
Conductor 728728
Sign −0.325+0.945i-0.325 + 0.945i
Analytic cond. 0.3633190.363319
Root an. cond. 0.6027590.602759
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(728s/2 ΓC(s) L(s)=((−0.325+0.945i) Λ‾(1−s)\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}
Λ(s)=(728s/2 ΓC(s) L(s)=((−0.325+0.945i) Λ‾(1−s)\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: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: −0.325+0.945i-0.325 + 0.945i
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ728(347,⋅)\chi_{728} (347, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 728, ( :0), −0.325+0.945i)(2,\ 728,\ (\ :0),\ -0.325 + 0.945i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.59036061570.5903606157
L(12)L(\frac12) ≈\approx 0.59036061570.5903606157
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+T 1 + T
7 1+iT 1 + iT
13 1−iT 1 - iT
good3 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
5 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1−2iT−T2 1 - 2iT - T^{2}
29 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
31 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
53 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
59 1+T2 1 + T^{2}
61 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
67 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line