↑

Properties

Label 1-667-667.592-r0-0-0
Degree $1$
Conductor $667$
Sign $0.446 - 0.894i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + (0.755 − 0.654i)2-s + (−0.540 + 0.841i)3-s + (0.142 − 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (−0.540 − 0.841i)8-s + (−0.415 − 0.909i)9-s + (−0.281 − 0.959i)10-s + (0.755 + 0.654i)11-s + (0.755 + 0.654i)12-s + (0.959 − 0.281i)13-s + (0.909 − 0.415i)14-s + (0.540 + 0.841i)15-s + (−0.959 − 0.281i)16-s + (−0.989 + 0.142i)17-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + (−0.540 + 0.841i)3-s + (0.142 − 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (−0.540 − 0.841i)8-s + (−0.415 − 0.909i)9-s + (−0.281 − 0.959i)10-s + (0.755 + 0.654i)11-s + (0.755 + 0.654i)12-s + (0.959 − 0.281i)13-s + (0.909 − 0.415i)14-s + (0.540 + 0.841i)15-s + (−0.959 − 0.281i)16-s + (−0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.446 - 0.894i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.446 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.900690792 - 1.176107809i\)
\(L(\frac12)\) \(\approx\) \(1.900690792 - 1.176107809i\)
\(L(1)\) \(\approx\) \(1.520145473 - 0.5511338321i\)
\(L(1)\) \(\approx\) \(1.520145473 - 0.5511338321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.755 - 0.654i)T \)
3 \( 1 + (-0.540 + 0.841i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (0.755 + 0.654i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.989 + 0.142i)T \)
31 \( 1 + (0.540 + 0.841i)T \)
37 \( 1 + (-0.909 + 0.415i)T \)
41 \( 1 + (-0.909 - 0.415i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (-0.540 - 0.841i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (0.281 + 0.959i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (0.540 - 0.841i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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

−22.84378659805969395712264086106, −22.41345987411173957629468652384, −21.59138988410904979326291799111, −20.7480556256679843024206306975, −19.60843552824464567707332474450, −18.45489109104658194613091428922, −17.90607927829984973802432546385, −17.25709975952564948572245408962, −16.40240168266872893288558182439, −15.38564583439535448071834570318, −14.3541973450721225806007200706, −13.74653196005610559055153947721, −13.369857612575998813848066456716, −11.93214718942600961839285717894, −11.35171140464806790921235399653, −10.804799261395527981889168370, −9.050176322446287338548933253697, −8.06024757527626482721375138672, −7.23464519659206356375546211909, −6.4495657037161784255308210440, −5.8737541814630182004168820814, −4.81690397062994040581363134095, −3.6797521970969627299615197559, −2.50957336905845907615743692982, −1.39545355585247156600932760283, 1.07449376696042805991071639788, 1.9523795083696939314267254729, 3.49993422563316827165627770767, 4.39632248669262880936533011539, 5.054476816650667938677035391481, 5.740004900066865489213743236604, 6.75315098591228765762799704495, 8.584319104605317412458391538791, 9.178178936938406606281190036157, 10.16888502219585602831231095086, 10.94762314185149563084140406942, 11.89355342217860906415536440739, 12.26045004724665669760045106916, 13.55090607357307822436517647307, 14.17883775136598413061904283090, 15.35155739208873047416180585665, 15.68466708570394161859157962157, 16.91986976700367262304105409555, 17.696194721988657374195235630084, 18.40049573380065835982801382498, 20.05028512221934482443664413251, 20.29560195333603429580083392649, 21.129003670860677273302891913946, 21.68522560548450032883978974399, 22.51808371673186347039178256150

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