↑

Properties

Label 1-980-980.843-r1-0-0
Degree $1$
Conductor $980$
Sign $-0.230 - 0.972i$
Analytic cond. $105.315$
Root an. cond. $105.315$
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  + (0.930 + 0.365i)3-s + (0.733 + 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.974 + 0.222i)13-s + (−0.997 + 0.0747i)17-s + (0.5 − 0.866i)19-s + (−0.997 − 0.0747i)23-s + (0.433 + 0.900i)27-s + (0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.930 − 0.365i)33-s + (−0.563 − 0.826i)37-s + (−0.988 − 0.149i)39-s + (−0.623 − 0.781i)41-s + (−0.781 − 0.623i)43-s + ⋯
L(s)  = 1  + (0.930 + 0.365i)3-s + (0.733 + 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.974 + 0.222i)13-s + (−0.997 + 0.0747i)17-s + (0.5 − 0.866i)19-s + (−0.997 − 0.0747i)23-s + (0.433 + 0.900i)27-s + (0.900 + 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.930 − 0.365i)33-s + (−0.563 − 0.826i)37-s + (−0.988 − 0.149i)39-s + (−0.623 − 0.781i)41-s + (−0.781 − 0.623i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.230 - 0.972i$
Analytic conductor: \(105.315\)
Root analytic conductor: \(105.315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (1:\ ),\ -0.230 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9978462607 - 1.262362563i\)
\(L(\frac12)\) \(\approx\) \(0.9978462607 - 1.262362563i\)
\(L(1)\) \(\approx\) \(1.266324184 - 0.04174902789i\)
\(L(1)\) \(\approx\) \(1.266324184 - 0.04174902789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.930 + 0.365i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + (-0.997 + 0.0747i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.997 - 0.0747i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.563 - 0.826i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (0.294 - 0.955i)T \)
53 \( 1 + (-0.563 + 0.826i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.294 - 0.955i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.974 - 0.222i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
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

−21.77938627126993539449052875409, −20.75058409391477756682453845829, −19.92903918100798996527610407197, −19.72104631104853890707967449119, −18.653374005451066603969184467208, −17.86903283000506267801520780392, −17.22397140042094554022559206829, −16.03970014607682654260908925681, −15.30899963816656960085281432492, −14.43967257845009479057575212927, −14.01913963923910383514017037669, −12.94309636840998319292155625495, −12.27654617088442143697315369692, −11.542007896227532297089528793693, −10.02499114738999615961576310785, −9.74055303957104433346555953608, −8.63589997952692340925076859774, −7.93701428548192179740532476284, −7.01555289016448224695044520924, −6.3805631420765627413306296906, −4.95459587817495341390916792207, −4.10042970602517868429040787894, −3.118449465278038803175486687219, −2.13115836835108357104617114648, −1.31453769941314115145603034313, 0.269484337894318349564336644034, 1.7915714721719999907362146509, 2.612434426740607086301094693719, 3.64818237049663054265720515191, 4.41578904018318648014576324243, 5.380185203806653090946879683005, 6.67807051227707150602137904217, 7.37002414796595951438097597940, 8.47189295332148572909286587217, 9.03790885144695827542670401117, 9.8236249387746292997258298548, 10.71597180494574011998328380509, 11.6575162762616050974991728040, 12.55170945523229069965730299130, 13.69435217759089568151058314823, 13.99603534863684386609014055942, 14.98182331681462584917971158505, 15.6354874818557585455499042044, 16.46626053697019585151456927905, 17.296069809132239847129744001286, 18.26674831106362073165990972522, 19.16546560125189041706046250523, 19.90935523921449609660131157668, 20.19275865661847398297943480632, 21.47895947222206715142510959580

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