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Properties

Label 2-980-140.59-c1-0-75
Degree $2$
Conductor $980$
Sign $0.799 + 0.601i$
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  + (−1.02 + 0.974i)2-s + (1.84 + 1.06i)3-s + (0.101 − 1.99i)4-s + (−1.92 − 1.13i)5-s + (−2.93 + 0.707i)6-s + (1.84 + 2.14i)8-s + (0.780 + 1.35i)9-s + (3.07 − 0.718i)10-s + (−2.01 − 1.16i)11-s + (2.32 − 3.58i)12-s + 1.09·13-s + (−2.35 − 4.15i)15-s + (−3.97 − 0.407i)16-s + (2.49 − 4.31i)17-s + (−2.11 − 0.625i)18-s + (−1.28 − 2.23i)19-s + ⋯
L(s)  = 1  + (−0.724 + 0.688i)2-s + (1.06 + 0.616i)3-s + (0.0509 − 0.998i)4-s + (−0.862 − 0.506i)5-s + (−1.19 + 0.288i)6-s + (0.650 + 0.759i)8-s + (0.260 + 0.450i)9-s + (0.973 − 0.227i)10-s + (−0.608 − 0.351i)11-s + (0.670 − 1.03i)12-s + 0.302·13-s + (−0.608 − 1.07i)15-s + (−0.994 − 0.101i)16-s + (0.604 − 1.04i)17-s + (−0.499 − 0.147i)18-s + (−0.295 − 0.511i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.799 + 0.601i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.799 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966751 - 0.323115i\)
\(L(\frac12)\) \(\approx\) \(0.966751 - 0.323115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.02 - 0.974i)T \)
5 \( 1 + (1.92 + 1.13i)T \)
7 \( 1 \)
good3 \( 1 + (-1.84 - 1.06i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.01 + 1.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.09T + 13T^{2} \)
17 \( 1 + (-2.49 + 4.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.28 + 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.02 + 5.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.561T + 29T^{2} \)
31 \( 1 + (3.29 - 5.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.76 + 2.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 + (-8.43 + 4.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.43 + 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.16 + 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.536 - 0.310i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.35 - 4.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.9iT - 71T^{2} \)
73 \( 1 + (-4.98 + 8.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.21 - 5.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.86iT - 83T^{2} \)
89 \( 1 + (-2.44 + 1.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−9.626087406321436302345555896333, −8.838842766723704545648278966678, −8.421082991430744491234320035204, −7.69633225903107415908816083627, −6.84278228684687726564304466736, −5.52685638609484169162041011891, −4.66182968937115525935617563283, −3.63455749417552107820089436083, −2.44720967476988265064525302127, −0.53568422400997397643373183340, 1.51341119999015367359328861715, 2.59592982894235727824295241420, 3.43705436522482984003611812336, 4.25804145156903205055698725281, 6.05943914776629856175419925407, 7.30236659869474188994511850359, 7.87740206398215885731482938405, 8.172188784369438262945279091755, 9.187655192859291746621067759270, 10.07916206191954090036065197224

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