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Properties

Label 2-980-28.19-c1-0-24
Degree $2$
Conductor $980$
Sign $-0.997 - 0.0661i$
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  + (−1.26 + 0.626i)2-s + (1.49 + 2.59i)3-s + (1.21 − 1.58i)4-s + (0.866 + 0.5i)5-s + (−3.52 − 2.35i)6-s + (−0.546 + 2.77i)8-s + (−2.99 + 5.18i)9-s + (−1.41 − 0.0915i)10-s + (−1.93 + 1.11i)11-s + (5.94 + 0.774i)12-s + 3.17i·13-s + 2.99i·15-s + (−1.04 − 3.86i)16-s + (2.98 − 1.72i)17-s + (0.548 − 8.45i)18-s + (−1.02 + 1.77i)19-s + ⋯
L(s)  = 1  + (−0.896 + 0.442i)2-s + (0.865 + 1.49i)3-s + (0.607 − 0.794i)4-s + (0.387 + 0.223i)5-s + (−1.43 − 0.960i)6-s + (−0.193 + 0.981i)8-s + (−0.998 + 1.72i)9-s + (−0.446 − 0.0289i)10-s + (−0.584 + 0.337i)11-s + (1.71 + 0.223i)12-s + 0.879i·13-s + 0.774i·15-s + (−0.261 − 0.965i)16-s + (0.723 − 0.417i)17-s + (0.129 − 1.99i)18-s + (−0.235 + 0.407i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.997 - 0.0661i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.997 - 0.0661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0415307 + 1.25389i\)
\(L(\frac12)\) \(\approx\) \(0.0415307 + 1.25389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.626i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (-1.49 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.17iT - 13T^{2} \)
17 \( 1 + (-2.98 + 1.72i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 - 1.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.30 + 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.38T + 29T^{2} \)
31 \( 1 + (-2.44 - 4.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.59 + 9.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 9.95iT - 43T^{2} \)
47 \( 1 + (3.06 - 5.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.32 + 4.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0456 - 0.0263i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.212iT - 71T^{2} \)
73 \( 1 + (-12.8 + 7.43i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.399 - 0.230i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (6.07 + 3.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.185iT - 97T^{2} \)
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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

−9.987296623294979579150235220896, −9.631631428670800153134728785154, −8.961798589884689184922409313669, −8.088693858042615801969920917762, −7.37002642283235167152235405244, −6.11882197541353967730184750742, −5.20495543921025010688870042465, −4.26267514361640666957703027488, −3.00554740320721776665386756877, −2.00356841190108365448475978284, 0.67242270674577860728007482688, 1.83899597315427692437709961449, 2.69392964552538853368720980538, 3.61494945956080402827018731111, 5.59000565686395762544604389079, 6.49564191613541507662759976930, 7.43053261115471269276006664250, 8.070810914948245347271470629649, 8.494069972947517754401801463213, 9.498010999719590744897860661645

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