↑

Properties

Label 1-920-920.597-r0-0-0
Degree $1$
Conductor $920$
Sign $0.850 + 0.525i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + i·3-s − i·7-s − 9-s + 11-s + i·13-s − i·17-s − 19-s + 21-s − i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + 41-s + ⋯
L(s)  = 1  + i·3-s − i·7-s − 9-s + 11-s + i·13-s − i·17-s − 19-s + 21-s − i·27-s + 29-s + 31-s + i·33-s + i·37-s − 39-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416859671 + 0.4025001407i\)
\(L(\frac12)\) \(\approx\) \(1.416859671 + 0.4025001407i\)
\(L(1)\) \(\approx\) \(1.080664673 + 0.2319258306i\)
\(L(1)\) \(\approx\) \(1.080664673 + 0.2319258306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 \)
89 \( 1 \)
97 \( 1 \)
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

−21.830799291550160493901522930112, −21.12932426146669753135672771514, −19.87383821357798111615759056409, −19.44559957889298199299298330994, −18.75661016046307702233980413544, −17.70084917129162842746171676932, −17.45102593040798285460613895399, −16.346016583938918236945619536424, −15.1993095848503361811801820290, −14.69094088362432189859311963101, −13.758372798137915874204001194927, −12.666737402105939533270967205338, −12.42503543300992292166319827557, −11.47550719829735869342272837169, −10.596238264572764007545744474082, −9.39545568556792050934593383718, −8.456729961747666169611972401211, −8.051277884049736431652799609227, −6.72028032495641697358741011804, −6.16538748365494012758348837347, −5.36307688041270743711997963256, −4.02754644210095855776692068874, −2.8409784093785592863856040502, −2.04742013476957890507594166240, −0.9299151724064702928509452738, 0.8933871029756798742277084039, 2.36889795078505857481400629295, 3.56496907552922844265523068003, 4.30436737815035557146390521283, 4.8836137766733654096542425942, 6.31110418377276262349660775870, 6.88777162903061075042430419707, 8.16139948940608991457269317827, 9.035592164116320624454332266068, 9.767132257095288526221996224701, 10.51333798932901248134136742046, 11.41326607960382908434511877192, 11.98207291457829328972413161143, 13.37609824683162059463274560144, 14.17429618450569685077036739929, 14.59417124908731204851575271230, 15.76080608348006801711024999257, 16.39475565038815486290621987872, 17.10703610107693768479054894573, 17.653233473509012366846867972945, 19.11858913451047251027753001689, 19.59614303292420576313943335921, 20.58505179677512565321652672085, 21.03523915044196071386866443493, 21.96811808526543088520841122966

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