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Properties

Label 2-980-28.27-c1-0-26
Degree $2$
Conductor $980$
Sign $0.725 + 0.688i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.976 − 1.02i)2-s − 1.11·3-s + (−0.0916 + 1.99i)4-s + i·5-s + (1.08 + 1.13i)6-s + (2.13 − 1.85i)8-s − 1.76·9-s + (1.02 − 0.976i)10-s − 1.71i·11-s + (0.101 − 2.22i)12-s − 2.45i·13-s − 1.11i·15-s + (−3.98 − 0.366i)16-s + 6.21i·17-s + (1.72 + 1.80i)18-s + 0.216·19-s + ⋯
L(s)  = 1  + (−0.690 − 0.723i)2-s − 0.642·3-s + (−0.0458 + 0.998i)4-s + 0.447i·5-s + (0.443 + 0.464i)6-s + (0.753 − 0.656i)8-s − 0.587·9-s + (0.323 − 0.308i)10-s − 0.516i·11-s + (0.0294 − 0.641i)12-s − 0.682i·13-s − 0.287i·15-s + (−0.995 − 0.0915i)16-s + 1.50i·17-s + (0.405 + 0.424i)18-s + 0.0496·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.725 + 0.688i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.725 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.676083 - 0.269857i\)
\(L(\frac12)\) \(\approx\) \(0.676083 - 0.269857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.976 + 1.02i)T \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 + 1.11T + 3T^{2} \)
11 \( 1 + 1.71iT - 11T^{2} \)
13 \( 1 + 2.45iT - 13T^{2} \)
17 \( 1 - 6.21iT - 17T^{2} \)
19 \( 1 - 0.216T + 19T^{2} \)
23 \( 1 + 6.56iT - 23T^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
31 \( 1 + 0.163T + 31T^{2} \)
37 \( 1 - 7.69T + 37T^{2} \)
41 \( 1 - 8.34iT - 41T^{2} \)
43 \( 1 + 1.89iT - 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 - 4.28T + 59T^{2} \)
61 \( 1 - 7.00iT - 61T^{2} \)
67 \( 1 + 5.17iT - 67T^{2} \)
71 \( 1 + 5.04iT - 71T^{2} \)
73 \( 1 + 7.61iT - 73T^{2} \)
79 \( 1 + 15.9iT - 79T^{2} \)
83 \( 1 - 5.47T + 83T^{2} \)
89 \( 1 + 1.78iT - 89T^{2} \)
97 \( 1 + 10.5iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.38539594375642733223045739467, −9.037954346340147061080255920562, −8.378695127186582378269080426837, −7.61035042892991348648639205167, −6.42621184441108168296708153430, −5.77392887461719931482281380318, −4.40155148710351467142479416817, −3.32801924900803614522287636659, −2.35094467339100054370109045033, −0.70135645215088026144002283737, 0.814977143771299697532100388009, 2.34656423077108842520966640066, 4.19995297559564467063223682289, 5.28479720196408038899267283925, 5.69119590849188556045558914300, 6.89876503348542375209896756499, 7.42298805762126748563729910630, 8.502311282169780382131056141130, 9.301442693751404294254735793320, 9.782300964512576161978367360000

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