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Properties

Label 1-864-864.205-r0-0-0
Degree $1$
Conductor $864$
Sign $0.964 + 0.265i$
Analytic cond. $4.01239$
Root an. cond. $4.01239$
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  + (0.996 − 0.0871i)5-s + (−0.342 + 0.939i)7-s + (−0.0871 + 0.996i)11-s + (0.819 − 0.573i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (0.573 − 0.819i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)35-s + (0.258 + 0.965i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (0.996 − 0.0871i)5-s + (−0.342 + 0.939i)7-s + (−0.0871 + 0.996i)11-s + (0.819 − 0.573i)13-s + (0.5 − 0.866i)17-s + (0.258 − 0.965i)19-s + (−0.342 − 0.939i)23-s + (0.984 − 0.173i)25-s + (0.573 − 0.819i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)35-s + (0.258 + 0.965i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(4.01239\)
Root analytic conductor: \(4.01239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 864,\ (0:\ ),\ 0.964 + 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.803800447 + 0.2441505423i\)
\(L(\frac12)\) \(\approx\) \(1.803800447 + 0.2441505423i\)
\(L(1)\) \(\approx\) \(1.296878872 + 0.09371753610i\)
\(L(1)\) \(\approx\) \(1.296878872 + 0.09371753610i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.996 - 0.0871i)T \)
7 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (-0.0871 + 0.996i)T \)
13 \( 1 + (0.819 - 0.573i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.573 - 0.819i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (0.984 + 0.173i)T \)
43 \( 1 + (-0.0871 + 0.996i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.996 - 0.0871i)T \)
61 \( 1 + (0.422 + 0.906i)T \)
67 \( 1 + (-0.819 + 0.573i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.573 + 0.819i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.84085440629209709749857411978, −21.29109434234872581977458446136, −20.57109747389293440064805978953, −19.61032155886614708597558725864, −18.80103062712700591713457512404, −18.06743957062319224796935776450, −17.10910522291164827527141837948, −16.52988145110374139804574188842, −15.85629988571582966054673083381, −14.37251412064146922453843933206, −14.042445064380197411027083816492, −13.25087432999963770540171276234, −12.485705522814100046262047132543, −11.16212164965381078174990915811, −10.59462145739377390210004410200, −9.77235649998700492306334538121, −8.92386395780013304959541196043, −7.94862026812815261199522901102, −6.91647264549508182728687962968, −6.00846982959666706973764716045, −5.47442188447160998465713539941, −3.9119126623398119641537698959, −3.40681235282972329795955884761, −1.94049723952575976050233836904, −1.04615363485606904254256874829, 1.09986075365189453187182618083, 2.385257017036272844487443306652, 2.916946099795347383326827126100, 4.474179815450682617911603718218, 5.35574685739586951680019852237, 6.105085032770438946096991980522, 6.94891851666980086714267135862, 8.12274308644554053823700292390, 9.108656994575595431906524602518, 9.665705463045328407891061836130, 10.49996461096942364303450248755, 11.57813255212321489665523958776, 12.53086299303007670358368641601, 13.09130668049269383212914829022, 14.00315693968640473714388566314, 14.87292839314548991179044700126, 15.71054926607245208963800666718, 16.40540636123619913153245260647, 17.545172102013838547813484943422, 18.091159956512087817191858935925, 18.64820961528127132232851922814, 19.83255823692371378954028217627, 20.64148333353661678951507189375, 21.20416893046263622684919079080, 22.24040673654064769862452251518

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