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Properties

Label 1-980-980.647-r1-0-0
Degree $1$
Conductor $980$
Sign $0.766 - 0.641i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.766 - 0.641i$
Analytic conductor: \(105.315\)
Root analytic conductor: \(105.315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (1:\ ),\ 0.766 - 0.641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.926935726 - 0.6997324013i\)
\(L(\frac12)\) \(\approx\) \(1.926935726 - 0.6997324013i\)
\(L(1)\) \(\approx\) \(1.020754626 - 0.1856010606i\)
\(L(1)\) \(\approx\) \(1.020754626 - 0.1856010606i\)

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 \( 1 - iT \)
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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

−21.54674210801308939041762821103, −21.00637374199584574315065040894, −20.18795745790475732143763521829, −19.11975245588176438095206603379, −18.3173053294632613212353891639, −17.68007521137442171428765608080, −16.72090086227006387140425493296, −16.33343566606415560896860449385, −15.343134608095554399956455343130, −14.59773740812847425196893329076, −13.66295309601141528679205948525, −12.5068447511186953180820832763, −12.04355284954322383008189474567, −11.16282834293349898424966827138, −10.36090531521958991576045070709, −9.61179717414555099756712464980, −8.75125662156853421740342331644, −7.53675221365885055035886804118, −6.669841304306866860827180552200, −5.89072929716749585404027905773, −5.04026182796053932074863144209, −4.071399767970052755961629403439, −3.31249556186961866818978198821, −1.6263396898482690943297202441, −0.84411540274275373197873585506, 0.77640917302826609312850080703, 1.26700577673444459599922383336, 2.82573291827244054941743657964, 3.85168269793390998839294155201, 4.97186304197344285301250348107, 5.77440838202813135518940064566, 6.518581729153463321686175920200, 7.35288863204490028414714470697, 8.342667658034320476548334237560, 9.2905257260535009495624058506, 10.303101103504224471120821977, 11.217409529572213201266284132799, 11.60951989153168928211922675950, 12.63381404013888306897414222313, 13.373265251430474067817216712426, 14.12351341504435206896895281624, 15.23162041410617373594473577666, 16.14452240261353512394686995961, 16.69917189358288185338769509416, 17.498697164777139735637844573949, 18.27822901509520209404830210098, 18.959688313641135237839650979283, 19.6802607750669008673053229260, 20.81839507374888511983231412260, 21.50399803414143489530489807110

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