↑

Properties

Label 1-1045-1045.753-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.883 - 0.468i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + i·12-s + (0.406 − 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.104 + 0.994i)24-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.207 + 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.104 + 0.994i)6-s + (−0.951 − 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.913 − 0.406i)9-s + i·12-s + (0.406 − 0.913i)13-s + (−0.978 − 0.207i)14-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.951 − 0.309i)18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (0.104 + 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.223660026 - 0.5533515944i\)
\(L(\frac12)\) \(\approx\) \(2.223660026 - 0.5533515944i\)
\(L(1)\) \(\approx\) \(1.649891301 + 0.007440660075i\)
\(L(1)\) \(\approx\) \(1.649891301 + 0.007440660075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.978 + 0.207i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.77632609296759141226153082421, −21.01095833492724160823187370048, −19.9113256871287071982391190633, −19.33349372551242868761758322402, −18.7483368088021508011857276381, −17.58763588063174965924480365310, −16.7782697066809112387964914487, −16.12786171978795379343924903958, −15.19661828248279863451781455345, −14.374850821910283306145321171587, −13.45590014344192248933723531749, −12.99685748522355961091861087309, −12.33419494740465644605886825682, −11.44999890328374364471732885007, −10.864261863064088125796312844959, −9.52461598069525986397868568794, −8.492943155249261304849773411375, −7.51058854855913831843115805223, −6.61152122401550493526368329991, −6.22130634709665055812741408716, −5.33341421756104216519608141128, −4.160274802416074554754975809059, −3.18473859733127335969979521411, −2.29474960604980383512731685021, −1.33385669679437249617288082498, 0.72389504520926273666244319634, 2.580713048979190993951376773068, 3.222574886451138266577299648552, 4.045080174917692674327463574621, 4.906072099950630226973172698888, 5.74125242717580751523754735287, 6.490572788407257838398819824385, 7.433702193522661100848613047099, 8.71186694032531856377210549195, 9.76240203249767805204405088752, 10.34967905978000810748971777995, 11.19621753477593450935421397462, 11.8548105967701419902611501678, 13.0718588138020404612484109547, 13.35306053825414433963173503850, 14.5814301111365478204329614777, 15.14969202093488198213684001037, 15.963262997178891239176449261282, 16.43875454485041474995016365372, 17.249054926265636494510172104945, 18.44611071642009993906014800417, 19.53194652143254863596015703741, 20.26755870317843543291530779629, 20.70849601726188590415885047438, 21.586379774812862741458942123930

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