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Properties

Label 1-980-980.719-r0-0-0
Degree $1$
Conductor $980$
Sign $0.315 - 0.949i$
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  + (0.988 − 0.149i)3-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.826 + 0.563i)23-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)33-s + (−0.0747 − 0.997i)37-s + (−0.365 − 0.930i)39-s + (−0.623 − 0.781i)41-s + (0.623 − 0.781i)43-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)3-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.826 − 0.563i)17-s + (−0.5 − 0.866i)19-s + (0.826 + 0.563i)23-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.988 − 0.149i)33-s + (−0.0747 − 0.997i)37-s + (−0.365 − 0.930i)39-s + (−0.623 − 0.781i)41-s + (0.623 − 0.781i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.315 - 0.949i$
Analytic conductor: \(4.55110\)
Root analytic conductor: \(4.55110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (0:\ ),\ 0.315 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559188373 - 1.125197657i\)
\(L(\frac12)\) \(\approx\) \(1.559188373 - 1.125197657i\)
\(L(1)\) \(\approx\) \(1.360055850 - 0.3384422846i\)
\(L(1)\) \(\approx\) \(1.360055850 - 0.3384422846i\)

Euler product

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

−21.64242601308817105164739014003, −20.8704816205719292081936636029, −20.579041883160922149800837692694, −19.40000814830745809881000881758, −18.83483768047099032618769023365, −18.284199820745152432350657150841, −16.89510345557578064256691792216, −16.41873241279070287347804401069, −15.32542974180325991413650918318, −14.73299016114201313316747265533, −14.100689171593257837593152192255, −13.009471539604223304047715522248, −12.631085241756688382087757447543, −11.37759826726045202406777113190, −10.371744395993216711240732551182, −9.74827998707672531789143550294, −8.85720793161654694715266701447, −8.00886193623428413956990743716, −7.37447712899713378510860840611, −6.316449942550041754803399397879, −5.11570838927621787779487806220, −4.236645807004923416808463463802, −3.33358914026588448624882304232, −2.34805683963446296107577954773, −1.49685714415014488757761388915, 0.722429272816432094520278310017, 2.12881803510103459031821788407, 2.93980216038037134722196207939, 3.68468162643593290751087457850, 4.97410433666815358466151666819, 5.71138899795610632590005104775, 7.22741927523027050067432339651, 7.52579842275296143509476118525, 8.59233445531823932035702901160, 9.24088944633316866254214022196, 10.25792409023430581497385119594, 10.89497299065591958792047550045, 12.19422620313063083453257418162, 12.95561505792315895058684664824, 13.55198903937034444544927757731, 14.3989187869598489041256959135, 15.33338585491369205262543074582, 15.68379547484318742491518846730, 16.83893944566787876242261200344, 17.797491637347204186129026380998, 18.564777146519869961845107623645, 19.21945904899819780934152135277, 20.01958538890533835676288058521, 20.76734957840864569340146830580, 21.32077656005820150049905154220

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