Properties

Label 1-980-980.867-r1-0-0
Degree $1$
Conductor $980$
Sign $-0.353 + 0.935i$
Analytic cond. $105.315$
Root an. cond. $105.315$
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  + (−0.974 + 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.781 − 0.623i)17-s − 19-s + (−0.781 + 0.623i)23-s + (−0.781 + 0.623i)27-s + (−0.623 + 0.781i)29-s + 31-s + (−0.974 − 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.781 − 0.623i)17-s − 19-s + (−0.781 + 0.623i)23-s + (−0.781 + 0.623i)27-s + (−0.623 + 0.781i)29-s + 31-s + (−0.974 − 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (0.222 + 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(105.315\)
Root analytic conductor: \(105.315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (1:\ ),\ -0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4426517512 + 0.6404287133i\)
\(L(\frac12)\) \(\approx\) \(0.4426517512 + 0.6404287133i\)
\(L(1)\) \(\approx\) \(0.7492619455 + 0.07585450218i\)
\(L(1)\) \(\approx\) \(0.7492619455 + 0.07585450218i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.433 - 0.900i)T \)
17 \( 1 + (-0.781 - 0.623i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + T \)
37 \( 1 + (0.781 + 0.623i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + T \)
83 \( 1 + (0.433 + 0.900i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 - iT \)
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

−21.58443647334118679317540083276, −20.59143513134584519379307374903, −19.41180843076335160156476119257, −18.99048791139166977418000456898, −18.060022184449685082086004862516, −17.229541943454611626052387066073, −16.71742658876895553973313862691, −15.93184211585665350659194636853, −15.01923821729782002387969145295, −14.01118002890787422608762804467, −13.23754202783911874578679376671, −12.35968008575774808331140239906, −11.59111033534633991210518145752, −10.98490967583230853523656068015, −10.13520535874640758114217256457, −9.06976017484313235625033234940, −8.266880343889965739960438110341, −7.065515606127751198199671515384, −6.29046510819537237196094228615, −5.8594433879230248670188377961, −4.2904542781403841310891567781, −4.13232852977753497970109906152, −2.31916352440926575702423927733, −1.4035425275165965018864602745, −0.23445392218046184721713900037, 0.893744522147596266578184356749, 1.98401514540610373080196408263, 3.44242929550618837036112731230, 4.34456469725009653877512568509, 5.13556362338028828407995704857, 6.1907353602329252125476497840, 6.71237095037328376764511479919, 7.79652652169140087200268141975, 8.887696424862818513401092777137, 9.786974985778571182387597285099, 10.526248062612171642158764232785, 11.39678258322795266778419052512, 11.998401834945684022480100130309, 12.91731191660741726788162793096, 13.63792295009875240430918753926, 14.96170741005488920875614055373, 15.37368116855519037624601303447, 16.4078942814471565316395328225, 17.000252163630515305149392053501, 17.903625433996530236804278682855, 18.25741570400956658428591461070, 19.50306566569468626389459160564, 20.19639304350583151705543189584, 21.07277134496499805354680152830, 21.99355280082554708072785798788

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