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Properties

Label 1-680-680.173-r0-0-0
Degree $1$
Conductor $680$
Sign $0.0253 - 0.999i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + 13-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 + 0.923i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s − i·33-s + (−0.382 + 0.923i)37-s + (0.382 − 0.923i)39-s + (−0.382 − 0.923i)41-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)7-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + 13-s + (0.707 − 0.707i)19-s − i·21-s + (0.382 + 0.923i)23-s + (−0.923 + 0.382i)27-s + (0.382 − 0.923i)29-s + (−0.923 − 0.382i)31-s − i·33-s + (−0.382 + 0.923i)37-s + (0.382 − 0.923i)39-s + (−0.382 − 0.923i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0253 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0253 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.0253 - 0.999i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.0253 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063573066 - 1.036935100i\)
\(L(\frac12)\) \(\approx\) \(1.063573066 - 1.036935100i\)
\(L(1)\) \(\approx\) \(1.077920334 - 0.4430141769i\)
\(L(1)\) \(\approx\) \(1.077920334 - 0.4430141769i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (-0.923 - 0.382i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.742608780744832249518335966447, −22.21440778913920059788253125167, −21.274970159684185357802201451375, −20.352317803089201964530787506544, −19.936303838592230037697627078493, −19.01530594141250968019275048484, −18.03551268393171464366371280007, −16.84579954335381338752644135315, −16.32654612087276461860003591074, −15.663407792889907201490890711096, −14.5658739719539669222701740275, −14.04147907926248811409694030629, −13.011553382300510333440476720001, −12.08003855268062612908954177800, −10.90502478591197141965217623791, −10.31381803671322232900432348153, −9.289686783539868745825671448053, −8.84882420072892008417307993917, −7.591854915587268033479812847887, −6.54483536801872912777999657078, −5.632702794369386338824777495480, −4.38115668622828886049553072642, −3.6686623580599469630208040359, −2.87402228761289969255235344940, −1.323042953007801768813863532602, 0.76835104855024536708920930405, 1.924724173457023108125348986514, 3.14518874483105415189860353247, 3.74802824680753046320237347142, 5.48088727947558180760999780427, 6.314138915378437597454618318721, 6.96353599101638842920356961947, 8.03075287290665801456339657140, 9.0506367921105496544651199409, 9.44865618375920429721643872715, 10.97149132570294016311452311437, 11.80146100659898169847713476317, 12.54059643542604815430560220509, 13.60424790048205353937281486723, 13.813197424373014932270059339367, 15.16118985505565088037789122029, 15.81511267668144892905162080818, 16.9248330389792241163601215070, 17.67686906856542522394897757475, 18.72282383717853777897464232524, 19.115937285805378663633964580854, 19.9555802195576511292502592288, 20.69157230451040246129273779097, 21.88482720704507862919100342932, 22.55125369020828264059036387487

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