↑

Properties

Label 1-975-975.17-r0-0-0
Degree $1$
Conductor $975$
Sign $-0.647 + 0.762i$
Analytic cond. $4.52788$
Root an. cond. $4.52788$
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  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 + 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.809 + 0.587i)14-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.978 − 0.207i)19-s + (−0.743 + 0.669i)22-s + (−0.406 + 0.913i)23-s + (−0.207 − 0.978i)28-s + (−0.978 + 0.207i)29-s + (−0.309 − 0.951i)31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.866 + 0.5i)7-s + (0.951 − 0.309i)8-s + (0.913 + 0.406i)11-s + (−0.809 + 0.587i)14-s + (−0.104 + 0.994i)16-s + (−0.207 + 0.978i)17-s + (−0.978 − 0.207i)19-s + (−0.743 + 0.669i)22-s + (−0.406 + 0.913i)23-s + (−0.207 − 0.978i)28-s + (−0.978 + 0.207i)29-s + (−0.309 − 0.951i)31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.647 + 0.762i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ -0.647 + 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4733677033 + 1.022931272i\)
\(L(\frac12)\) \(\approx\) \(0.4733677033 + 1.022931272i\)
\(L(1)\) \(\approx\) \(0.7548759922 + 0.4992717128i\)
\(L(1)\) \(\approx\) \(0.7548759922 + 0.4992717128i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (-0.207 + 0.978i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (-0.406 + 0.913i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (-0.913 - 0.406i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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

−21.292593662931972461739918772586, −20.54052508818572779175496770696, −20.01844354063579250730022018919, −19.11248171915881633245156456225, −18.39371391716818342636100536435, −17.6150622177319425575818632561, −16.90165820000567668246471999866, −16.27808394919918120192628683912, −14.8508981232149912610162000041, −14.130631299024140809957245311621, −13.47616642763362419949954912527, −12.40153243097925306402259800560, −11.72431265445575979605959243242, −10.925816255799889360054054854719, −10.37276766943647991775715082549, −9.19107610086319363785075041958, −8.66430083832528836018127646545, −7.69854924682440926164914532144, −6.86327733701472074811870741135, −5.515994305039359279669952698141, −4.36628949190706239684805781255, −3.890268848146415694241680262758, −2.59042655416429660421943195709, −1.68903744473927455500604312051, −0.60966072981779804805484268857, 1.316064926576808206613165427352, 2.13542184220323732773054158381, 3.960806615071649440895685782928, 4.56891990675799155153460045765, 5.74371206638489305048866064791, 6.276199301610624495954121024196, 7.42466116043714782364080954150, 8.070064102695879160284818730850, 8.986108495601188403130262810327, 9.54538243364565901618305541909, 10.71669505633163036610765933688, 11.42263791451118444835364897597, 12.53933650515787838618103838350, 13.41597658368450388860913404355, 14.45173880735094503625055706937, 14.95879471443614737331129419189, 15.49633515351348137675867704753, 16.75929773409611335183727234630, 17.18309303556694236353504029051, 17.948736193651921290879927604762, 18.68911502338869986911089039258, 19.53607395823244846496261706602, 20.22112744109157450235417030208, 21.47727297662550539898173691084, 22.00535819382907965058166440854

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