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Properties

Label 1-980-980.403-r0-0-0
Degree $1$
Conductor $980$
Sign $-0.161 + 0.986i$
Analytic cond. $4.55110$
Root an. cond. $4.55110$
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  + (−0.294 − 0.955i)3-s + (−0.826 + 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.433 − 0.900i)13-s + (−0.930 − 0.365i)17-s + (−0.5 + 0.866i)19-s + (0.930 − 0.365i)23-s + (0.781 + 0.623i)27-s + (−0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.294 + 0.955i)33-s + (0.149 + 0.988i)37-s + (−0.733 + 0.680i)39-s + (−0.222 + 0.974i)41-s + (−0.974 + 0.222i)43-s + ⋯
L(s)  = 1  + (−0.294 − 0.955i)3-s + (−0.826 + 0.563i)9-s + (−0.826 − 0.563i)11-s + (−0.433 − 0.900i)13-s + (−0.930 − 0.365i)17-s + (−0.5 + 0.866i)19-s + (0.930 − 0.365i)23-s + (0.781 + 0.623i)27-s + (−0.623 − 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.294 + 0.955i)33-s + (0.149 + 0.988i)37-s + (−0.733 + 0.680i)39-s + (−0.222 + 0.974i)41-s + (−0.974 + 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(4.55110\)
Root analytic conductor: \(4.55110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (0:\ ),\ -0.161 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09869327152 + 0.1161754731i\)
\(L(\frac12)\) \(\approx\) \(0.09869327152 + 0.1161754731i\)
\(L(1)\) \(\approx\) \(0.6549759562 - 0.2071329447i\)
\(L(1)\) \(\approx\) \(0.6549759562 - 0.2071329447i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.294 - 0.955i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (-0.433 - 0.900i)T \)
17 \( 1 + (-0.930 - 0.365i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.930 - 0.365i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.149 + 0.988i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.974 + 0.222i)T \)
47 \( 1 + (0.997 + 0.0747i)T \)
53 \( 1 + (0.149 - 0.988i)T \)
59 \( 1 + (-0.733 + 0.680i)T \)
61 \( 1 + (-0.988 + 0.149i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.997 + 0.0747i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.433 + 0.900i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 - iT \)
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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

−21.683375465840346510865333381206, −20.75739058227478450064487775304, −20.11993233030837909566626692585, −19.22333427159142390415460899150, −18.276450223584009895017058918590, −17.28462908476759072753062848662, −16.93395972308070274759627519512, −15.7869276993860018028078973881, −15.32553917329529631520920179317, −14.597307349843861929580039003421, −13.535777674175852116252526899827, −12.70191855550901593242675963532, −11.70140343438620928149311176608, −10.92671039715188220265933043638, −10.32748214129364574111518523728, −9.224901089901176765045137099661, −8.871047984386520803209807463251, −7.49814146183994018748092488413, −6.67139249002764270165830217017, −5.59405759688936414607789648268, −4.72701694230893742334282294688, −4.15001661286621965631879814313, −2.92063648719273688853507561596, −1.97537646301273859393549499130, −0.06910673032459427598561485153, 1.19990551571889155334725066330, 2.43768948568477535703613341283, 3.111732307675719099659072792962, 4.65640604349183952860143609852, 5.48849605036830006952605010477, 6.31845067903721479325008593923, 7.18200704226097462961745550267, 8.08444502638800238278898735396, 8.60929308966065132560489016926, 9.97691343095671927051237578119, 10.7993358281898069125127853984, 11.53336511900157978668767570173, 12.48984621067958337313935342842, 13.138895182274115418364545396768, 13.72290552461073337849702321346, 14.80510024963048608133746826748, 15.573386312365455597152981654012, 16.65655944989420421576981028891, 17.22983592186749383335428131674, 18.17967472963954189771162676836, 18.640127518539207196377646419061, 19.49492538865166125672728176109, 20.251655463662484024992993140638, 21.09879766624576982028215628759, 22.07915474726703830024337515266

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