↑

Properties

Label 1-2960-2960.197-r1-0-0
Degree $1$
Conductor $2960$
Sign $-0.567 - 0.823i$
Analytic cond. $318.096$
Root an. cond. $318.096$
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  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)7-s + (−0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.939 + 0.342i)13-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (−0.642 + 0.766i)21-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s + 31-s + (−0.342 + 0.939i)33-s + (0.173 − 0.984i)39-s + (0.939 + 0.342i)41-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)7-s + (−0.939 + 0.342i)9-s + (−0.866 − 0.5i)11-s + (0.939 + 0.342i)13-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (−0.642 + 0.766i)21-s + (0.866 − 0.5i)23-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s + 31-s + (−0.342 + 0.939i)33-s + (0.173 − 0.984i)39-s + (0.939 + 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $-0.567 - 0.823i$
Analytic conductor: \(318.096\)
Root analytic conductor: \(318.096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2960,\ (1:\ ),\ -0.567 - 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7718729479 - 1.469129083i\)
\(L(\frac12)\) \(\approx\) \(0.7718729479 - 1.469129083i\)
\(L(1)\) \(\approx\) \(0.8013976035 - 0.4366085215i\)
\(L(1)\) \(\approx\) \(0.8013976035 - 0.4366085215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.642 - 0.766i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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

−19.184427282979766549173431475186, −18.49610982919866954705292489626, −17.56571460987111794386716978805, −17.15150089495935081070743617145, −16.06821952157412045428060507412, −15.72196129923863655208419602127, −15.12450413185196191767750587467, −14.56954856851333514070546810928, −13.29882295741320127232619789452, −12.90715062255952386424585035693, −12.09586449076181264082437740585, −11.09466229102770450044253916078, −10.6499549401879338665230735908, −9.96891292817830412656722720679, −9.03511118001410168091160571920, −8.69951235756384290177748102530, −7.755624255047584582105743966167, −6.61195892020262779260055501251, −5.88850177502250509498872802077, −5.36359833499071124977664857183, −4.4036520603240686679338216172, −3.68374521987089021774000114079, −2.84828604989840412453472196608, −2.10291754688811841325936812827, −0.64985608103385823415274250646, 0.4800221125239521820240809409, 0.920270978389974927500054136898, 2.22224689093689089130746984365, 2.843173124893936629483573359776, 3.82371285435123361450531830448, 4.73322627325929461121185126311, 5.80271333422745338302510615960, 6.33100261132248523387439578460, 7.08789827497296297920275474084, 7.70641860452018677806295643531, 8.554998647256334693298647383392, 9.19233108246635600798335235092, 10.32961614811573186960407778291, 10.94898258564615143305262355825, 11.48735144997359761913645148652, 12.55111011893550865729591396541, 13.0822572562145945692039523948, 13.62806033676240232365985096904, 14.128265627488852033126240256125, 15.23296663786579778285516262838, 16.04620028554672361804478129632, 16.65681150793067257222151152706, 17.26788956921289705678344485325, 18.0973040943925717920037015156, 18.82684985733437472641629630338

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