L-function 1-864-864.205-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
good	5	$1 + (0.996 - 0.0871i)T$
	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	$1 + (-0.939 + 0.342i)T$

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