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Properties

Label 2-980-140.139-c1-0-28
Degree $2$
Conductor $980$
Sign $0.654 - 0.755i$
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  − 1.41·2-s − 0.356i·3-s + 2.00·4-s + 2.23i·5-s + 0.504i·6-s − 2.82·8-s + 2.87·9-s − 3.16i·10-s − 0.712i·12-s + 0.796·15-s + 4.00·16-s − 4.06·18-s + 4.47i·20-s + 7.50·23-s + 1.00i·24-s − 5.00·25-s + ⋯
L(s)  = 1  − 1.00·2-s − 0.205i·3-s + 1.00·4-s + 0.999i·5-s + 0.205i·6-s − 1.00·8-s + 0.957·9-s − 1.00i·10-s − 0.205i·12-s + 0.205·15-s + 1.00·16-s − 0.957·18-s + 1.00i·20-s + 1.56·23-s + 0.205i·24-s − 1.00·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985664 + 0.450301i\)
\(L(\frac12)\) \(\approx\) \(0.985664 + 0.450301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 - 2.23iT \)
7 \( 1 \)
good3 \( 1 + 0.356iT - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 7.50T + 23T^{2} \)
29 \( 1 - 4.74T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 12.6iT - 41T^{2} \)
43 \( 1 + 9.10T + 43T^{2} \)
47 \( 1 - 9.48iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.6iT - 61T^{2} \)
67 \( 1 - 15.8T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 18.2iT - 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.938268332633974475939763790757, −9.545416994927608973390286431584, −8.331070747724469977314419557315, −7.63506187359729314646471480111, −6.74881306272918346306901123491, −6.41976545061979647550339160938, −4.94371696454789572029952107645, −3.47237065573252395276696544878, −2.52191379702123190476414787421, −1.23710565069708736530984518122, 0.817856809515735631835589491243, 1.95731489135935979144248582017, 3.45899190151610146228060182281, 4.67448197264196704482487757674, 5.57222428930692081750972821114, 6.80930669043799479232841035917, 7.41495333826327862510420186312, 8.548981937431419698628108282446, 8.920830983327324690986537237439, 9.890959292944921151305720344293

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