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Properties

Label 1-24e2-576.101-r1-0-0
Degree $1$
Conductor $576$
Sign $0.564 + 0.825i$
Analytic cond. $61.8997$
Root an. cond. $61.8997$
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  + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (0.130 − 0.991i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.258 − 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (0.923 + 0.382i)37-s + (−0.258 − 0.965i)41-s + (0.991 − 0.130i)43-s + (0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (0.130 − 0.991i)13-s − i·17-s + (0.923 − 0.382i)19-s + (−0.258 − 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (0.923 + 0.382i)37-s + (−0.258 − 0.965i)41-s + (0.991 − 0.130i)43-s + (0.866 − 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(61.8997\)
Root analytic conductor: \(61.8997\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 576,\ (1:\ ),\ 0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.531226828 + 0.8077512962i\)
\(L(\frac12)\) \(\approx\) \(1.531226828 + 0.8077512962i\)
\(L(1)\) \(\approx\) \(1.000933977 + 0.2018904931i\)
\(L(1)\) \(\approx\) \(1.000933977 + 0.2018904931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
13 \( 1 + (0.130 - 0.991i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.258 - 0.965i)T \)
43 \( 1 + (0.991 - 0.130i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.793 + 0.608i)T \)
61 \( 1 + (-0.608 + 0.793i)T \)
67 \( 1 + (0.991 + 0.130i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.793 + 0.608i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−23.02900042020428899339660546010, −22.21460335351823019068571825311, −21.045684748246678477499876609927, −20.27940449966369294920381667457, −19.66197910512011749504278247318, −18.94997872118869163501181873199, −17.769229246749913061768492425994, −16.80160377564678325448458100084, −16.30543974089761211410750388223, −15.48931184947560909061867282755, −14.19978908632005635310093364926, −13.73398199079946812894539048879, −12.56864980343787489206736352574, −11.67629789869277205154150432946, −11.17448929631234950910191116315, −9.63948970562134467244877773131, −9.26312891342350889007553660655, −7.89204165257114550951532696532, −7.26367525739468106465649847364, −6.280807423466272999843334802833, −4.88005575507231856550747668588, −4.08961485851749134251818932791, −3.325114177399173243634746887261, −1.55127760412484844976663299708, −0.61965564484770463744206418111, 0.828909068808580586697089204255, 2.4202115691306106711371582920, 3.33330659087231054284979764900, 4.242157269808882430342632957511, 5.63822269966013806159804246778, 6.38284603334242074173872125187, 7.457265574387029881281551682997, 8.380005884298618515016274479294, 9.203325058133426984347972124227, 10.35062774148931350098424816505, 11.209981025698691980725018469465, 12.07956309096674560260104335521, 12.69989418909356039583456936420, 13.97096448333649827075759131706, 14.951866798083613233215146821785, 15.36961348728606625611575819894, 16.31021248074468803299486200416, 17.331888985472548073350541838818, 18.35795228057911876322491834377, 18.89988641293878071854964867504, 19.808383503217585524569071182442, 20.437287460064102394130882192930, 21.886565908139085491285165368590, 22.2386497629100774250985988548, 22.93531719051291180421631434909

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