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Properties

Label 1-980-980.59-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  + (−0.365 + 0.930i)3-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.365 + 0.930i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)39-s + (−0.623 − 0.781i)41-s + (0.623 − 0.781i)43-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.0747 + 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)23-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (−0.5 − 0.866i)31-s + (0.365 + 0.930i)33-s + (−0.826 + 0.563i)37-s + (0.988 + 0.149i)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: $\chi_{980} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (0:\ ),\ 0.315 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6032293642 - 0.4353240950i\)
\(L(\frac12)\) \(\approx\) \(0.6032293642 - 0.4353240950i\)
\(L(1)\) \(\approx\) \(0.8117178958 + 0.05626548935i\)
\(L(1)\) \(\approx\) \(0.8117178958 + 0.05626548935i\)

Euler product

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

−21.986818179604599725110327948319, −21.166682875074831592724217688405, −19.957145703089370168262226238322, −19.59542358855701141560573904582, −18.67473206939361613734647715576, −17.95294706658515335225516848030, −17.23115397360045870112837696551, −16.58119965644956288995828822270, −15.60722610132035177176697856959, −14.49118741377207131201990270395, −13.945547826869038057732859406675, −13.014927696292327508850391124264, −12.271767691458796230416490588867, −11.519251732940011367583911670516, −10.92938365438017998454419872154, −9.50827722603587704836647381716, −9.01659915749379736091445076684, −7.74899698249250646685545565886, −7.010385988029605401997632674411, −6.503565638467938597973416831768, −5.29842606335845414963553285290, −4.54276867800242116557159787522, −3.2313257600668465593835443681, −2.06777167961875391084798749440, −1.32010308230036976718180661400, 0.33874176442992467084672623086, 1.89119969314154308465968520584, 3.3325878626843310383032345306, 3.85397240884171583929151632912, 4.92488236266192697927683307586, 5.8780527547737921727090452522, 6.40492491064844974040506710980, 7.86478246790600655286051172982, 8.62786340820737853251524515259, 9.470918773091230601101882060815, 10.46073928741325172425529114819, 10.83637679348220391474631292314, 11.93279295163621316302979565737, 12.59409068636610088758494352125, 13.705753764974748468500798698982, 14.76492847631863597578324624228, 15.07030492826294464048664379677, 16.11717581977147440038502108849, 16.98453791504731010086782842089, 17.20457311557004009445966925985, 18.44609648964742620302530188799, 19.23457624914066705014561629443, 20.185375976020325955992764997894, 20.818872157963068008125700421970, 21.589280881108114947994271007553

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