↑

Properties

Label 1-2240-2240.67-r0-0-0
Degree $1$
Conductor $2240$
Sign $-0.966 - 0.258i$
Analytic cond. $10.4025$
Root an. cond. $10.4025$
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  + (−0.991 + 0.130i)3-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)13-s + (0.5 + 0.866i)17-s + (−0.608 − 0.793i)19-s + (0.258 + 0.965i)23-s + (−0.923 + 0.382i)27-s + (−0.923 − 0.382i)29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (−0.130 + 0.991i)37-s + (0.965 + 0.258i)39-s + (−0.707 − 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)3-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)11-s + (−0.923 − 0.382i)13-s + (0.5 + 0.866i)17-s + (−0.608 − 0.793i)19-s + (0.258 + 0.965i)23-s + (−0.923 + 0.382i)27-s + (−0.923 − 0.382i)29-s + (−0.5 − 0.866i)31-s + (−0.866 − 0.5i)33-s + (−0.130 + 0.991i)37-s + (0.965 + 0.258i)39-s + (−0.707 − 0.707i)41-s + (−0.382 − 0.923i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.966 - 0.258i$
Analytic conductor: \(10.4025\)
Root analytic conductor: \(10.4025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2240,\ (0:\ ),\ -0.966 - 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0001462858294 + 0.001114055201i\)
\(L(\frac12)\) \(\approx\) \(0.0001462858294 + 0.001114055201i\)
\(L(1)\) \(\approx\) \(0.6556670833 + 0.04673839055i\)
\(L(1)\) \(\approx\) \(0.6556670833 + 0.04673839055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (0.793 + 0.608i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.608 - 0.793i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.130 + 0.991i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.608 + 0.793i)T \)
59 \( 1 + (0.608 - 0.793i)T \)
61 \( 1 + (0.793 - 0.608i)T \)
67 \( 1 + (0.991 - 0.130i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.965 - 0.258i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (0.258 + 0.965i)T \)
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

−19.14849786580017045644138048712, −18.57090631167440380888040468010, −17.86701661410227816684123634940, −16.999302227767707555964588072419, −16.47750295505426878730349721737, −16.13147643663711455281965148507, −14.65748198391926981120360130927, −14.546951562958449014900665140152, −13.341623433534415815804083319942, −12.61487531101644201190123915302, −11.930014257069179137768224822220, −11.402208847137099813991797264666, −10.55966426568390075903213879963, −9.84821549380885430505349572161, −9.0615873883243732863337058565, −8.10066086323858197985979633572, −7.09156478614931114100004675747, −6.66071080263313365808589017284, −5.71453963069749212960677552413, −5.04656703939476352489950612996, −4.22097858765366694928175412761, −3.31085608392905391887149380651, −2.081809056893581236115450049449, −1.1935400763954768710398450033, −0.00046127267703606987255222156, 1.32645168821431999594284166268, 2.1479842197812799125827632215, 3.5085770746818046893091394626, 4.24495677220249759875626494651, 5.08771613858385170250651918013, 5.74405327703918019047272152992, 6.667637168930118916432701158075, 7.235966989794522968598966118927, 8.11745193037528814323071316790, 9.335269885187883260366792298562, 9.791256578168279794428238711118, 10.61282206719421055691308861087, 11.39309768132331484808099593454, 12.00532006999342237677967990415, 12.735510723773438523675284156897, 13.30530663760944343460262954409, 14.53969587279706356052001746559, 15.13295804277768954036510184705, 15.67510952768140047632467217010, 16.84596423207542373515756379967, 17.236789959749436914500976057607, 17.54167721903782663835366041390, 18.7438846427452336138023020324, 19.19220064042855371465199349109, 20.12166243279487590643735102770

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